# Matematički blogovi

### Wolpert’s examples of tiny Weil-Petersson sectional curvatures revisited

In this previous post here (from 2018), I described some “back of the envelope calculations” (based on private conversations with Scott Wolpert) indicating that some sectional curvatures of the Weil–Petersson (WP) metric could be at least exponentially small in terms of the distance to the boundary divisor of Deligne–Mumford compactification.

Very roughly speaking, this heuristic computation went as follows: the WP sectional curvature of any -plane can be written as the sum of three terms; for the -planes considered in the previous post, the main term among those three *seemed* to be a kind of -norm of Beltrami differentials with essentially disjoint supports; finally, this -type norm was shown to be really small once a certain Green propagator is *ignored*.

Last April 2019, I met Scott during an event at Simons Center for Geometry and Physics, and I took the opportunity to tell him that one could perhaps show that the measure of the set of -planes leading to tiny WP curvatures is very small using the real-analyticity of the WP metric.

More concretely, my idea was very simple: since the Grassmannian of -planes tangent to a point is a compact space, the WP sectional curvature defines a real-analytic function , and we dispose of good upper bounds for and all of its derivatives in terms of the distance of to the boundary (see this article here), we can hope to get reasonable estimates for the measure of the sets using the techniques of these articles here and here (which are close in spirit to the classical fact [explained in Lemma 3.2 of Kleinbock–Margulis paper, for instance] that the measure of the sets are small whenever is a polynomial function on whose degree and -norm are bounded).

As it turns out, Scott thought that this strategy made some sense and, in particular, he promised to use my suggestion as a motivation to review his arguments concerning WP sectional curvatures.

After several email exchanges with Howard Masur and I, Scott announced that there were some *mistakes* in the construction of tiny WP sectional curvature: in a nutshell, one should not restrict the analysis to a single “main term” in the formula for WP sectional curvatures as a sum of three expressions, and one can not ignore the effect of the Green propagator. More importantly, Scott made a detailed study of these mistakes which ultimately led him to establish polynomial upper bounds for WP sectional curvatures at the heart of his newest preprint available here.

In this post, we will follow closely Scott’s preprint in order to give an outline of the proof of a polynomial upper bound for WP sectional curvatures:

**Theorem 1 (Wolpert)** *Given two integers and with , there exists a constant with the following property.*If denotes the product of the lengths of the short geodesics of a hyperbolic surface of genus with cusps whose systole is sufficiently small, then the sectional curvatures of the Weil-Petersson metric at are at most

**Remark 1** *As it was pointed out by Scott in his preprint, it is likely that this estimate is not optimal: indeed, one expects that the best exponent should be rather than .*

In what follows, we’ll assume some familiarity with some basic aspects of the geometry of the Weil–Petersson metric (such as those described in these posts here and here).

**1. Weil–Petersson sectional curvatures**

Let be a hyperbolic surface of genus with . If we write , where is the usual hyperbolic plane and is a group of isometries of describing the fundamental group of , then the holomorphic *tangent* space at to the moduli space of Riemann surfaces of genus with punctures is naturally identified with the space of harmonic Beltrami differentials on (and the *cotangent* space is related to quadratic differentials).

In this setting, the Weil–Petersson metric is the Riemannian metric induced by the Hermitian inner product

where and is the hyperbolic area form on .

**Remark 2** *Note that is well-defined: if and are Beltrami differentials, then is a function on .*

The Riemann tensor of the Weil–Petersson metric was computed by Wolpert in 1986:

where and is an operator related to the Laplace–Beltrami operator on .

**Remark 3** *Our choice of notation here differs from Wolpert’s preprint! Indeed, he denotes the Laplace–Beltrami operator by and he writes .*

The Riemann tensor gives access to nice formulas for the sectional curvatures thanks to the work of Bochner. More concretely, given and span a -plane in the real tangent space to at , let us take Beltrami differentials and such that , , and is orthonormal. Then, Bochner showed that the sectional curvature of is

Hence, by Wolpert’s formula for the Riemann tensor of the WP metric, we see that

**2. Spectral theory of **

Wolpert’s formula for the Riemann tensor of the WP metric hints that the spectral theory of plays an important role in the study of the WP sectional curvatures.

For this reason, let us review some key properties of (and we refer to Section 3 of Wolpert’s preprint for more details and references). First, is a positive operator on whose norm is : these facts follow by integration by parts. Secondly, is essentially self-adjoint on , so that is self-adjoint on . Moreover, the maximum principle permits to show that is also a positive operator on with unit norm. Finally, has a positive symmetric integral kernel: indeed,

where the Green propagator is the Poincaré series

associated to an appropriate Legendre function . (Here, stands for the hyperbolic distance on .) For later reference, we recall that has a logarithmic singularity at and whenever is large.

**3. Negativity of the WP sectional curvatures**

Interestingly enough, as it was first noticed by Wolpert in 1986, the spectral features of described above are sufficient to derive the negativity of WP sectional curvatures from Cauchy-Schwarz inequality. More precisely, since is self-adjoint, i.e.,

and its integral kernel is a *real* function, a straightforward computation reveals that the equation (1) for the sectional curvature of a -plane can be rewritten as

If we decompose the function into its real and imaginary parts, say , then we see that

Since is a positive operator, we conclude that and, *a fortiori*,

The non-positivity of the right-hand side of (2) can be established in three steps. First, the positivity of also implies that

Secondly, the fact that has a positive integral kernel allows to apply the Cauchy–Schwarz inequality to get that . Therefore,

Finally, the Cauchy–Schwarz inequality also says that

In particular, , so that it follows from (2) that all sectional curvatures of the WP metric are non-positive, i.e., .

Actually, it is not hard to derive that at this stage: indeed, would force a case of equality in Cauchy-Schwarz inequality and this is not possible in our context because is orthonormal.

**Remark 4** *Philosophically speaking, the “analog” to this argument in the realm of Teichmüller dynamics is Forni’s proof of the spectral gap property for the Lyapunov exponents of the Teichmüller geodesic flow. In fact, after some computations with variational formulas for the so-called Hodge norm, Forni establishes that by ruling out an equality case in a certain Cauchy-Schwarz estimate.*

**4. Reduction of Theorem 1 to bounds on ‘s kernel**

The discussion in the previous section says that small WP sectional curvatures correspond to almost equalities in certain Cauchy-Schwarz inequalities.

Hence, a natural strategy towards the proof of Theorem 1 consists into showing that an almost equality in (3) is impossible. In this direction, Wolpert establishes the following result:

**Theorem 2 (Wolpert)** *There are two constants and with the following property. If we have an almost equality*

*between the terms and in (3), then and can not be almost equal:*

Of course, Theorem 1 is an immediate consequence of Theorem 2 (in view of (2) and the estimate [implied by (3)]).

Thus, it remains only to prove Theorem 2. For this sake, we need further spectral information on , namely, some lower bounds on its the kernel . In order to illustrate this point, let us now show Theorem 2 *assuming* the following statement.

**Proposition 3** *There exists a constant such that*

*whenever and do not belong to the cusp region of .*

**Remark 5** *We recall that the cusp region of is a finite union of portions of which are isometric to a punctured disk (equipped with the hyperbolic metric ).*

For the sake of exposition, let us first establish Theorem 2 when is compact, i.e., , before explaining the extra ingredient needed to treat the general case.

**4.1. Proof of Theorem 2 modulo Proposition 3 when **

Suppose that for a constant to be chosen later. In this regime, our goal is to show that is “big” and is “small”, so that is necessarily “big”.

We start by quickly showing that is “big”. Since and are unitary tangent vectors, it follows from Proposition 3 that

Let us now focus on proving that is “small”. Since (cf. (3)), if we write (where and are the positive and negative parts of the real part of , then we obtain that

Since is positive, we derive that . Thus, if is compact, i.e., , then Proposition 3 says that for all . It follows that

By orthogonality of , we have that , i.e., . By plugging this information into the previous inequality, we obtain the estimate

Next, we observe that (cf. (3)) in order to obtain that

On the other hand, Proposition 3 ensures that for . Since and are unitary tangent vectors, one has for . By inserting this inequality into the previous estimate, we derive that

whenever has a sufficiently small systole.

This bound on can be converted into a bound thanks to Cauchy integral formula. More concrentely, as it is explained in Section 2 of Wolpert’s preprint, after observing that and replacing Beltrami differentials and by the dual objects and (namely, quadratic differentials), we are led to study quartic differentials . By Cauchy integral formula on , one has

On the other hand, if has systole and the cusp region is empty, then the injectivity radius at any is . Thus, there exists an universal constant such that

for all . By plugging this inequality into (7), we conclude that

for all .

Since is a positive operator on with unit norm (cf. Section 2 above) and , we have that the previous inequality implies the following bound on :

for all . By combining this estimate with (7), we conclude that

In summary, (4) and (8) imply that

for the choice of constant . This proves Theorem 2 in the absence of cusp regions.

**4.2. Proof of Theorem 2 modulo Proposition 3 when **

The arguments above for the case also work in the case because the cusp regions carry only a tiny fraction of the mass of the relevant functions, Beltrami differentials, etc.

More precisely, as it is explained in Section 2 of Wolpert’s preprint, if the constant is chosen correctly, then the Cauchy integral formula and the Schwarz lemma can be used to prove that

for all holomorphic quartic differentials .

In particular, we do not lose too much information after truncating , , etc. to and this allows us to repeat the arguments of the case to the corresponding truncated objects , , etc. without any extra difficulty: see Section 5 of Wolpert’s preprint for more details.

**5. Proof of Proposition 3**

Closing this post, let us give an idea of the proof of Proposition 3 (and we refer the reader to Section 4 of Wolpert’s preprint for more details).

Since and (cf. Section 2 above), our task is reduced to give lower bounds on the Poincaré series

For this sake, let us first recall that a hyperbolic surface has thick-thin decomposition: the thick portion is the region where the injectivity radius is bounded away from zero by a uniform constant and the thin portion is the complement of the thick region. Geometrically, the thin region is the disjoint union of the cusp region and a finite number of *collars* around simple closed short geodesics: roughly speaking, a collar consisting of the points at distance of a short simple closed geodesic of length .

We can provide lower bounds on in terms of the behaviours of simple geodesic arcs connecting and on .

More concretely, let be the shortest geodesic connecting and . Since is simple, we have that, for certain adequate choices of the constants defining the collars, one has that can not “back track” after entering a collar, i.e., it must connect the boundaries (rather than going out via the same boundary component). Furthermore, can not go very high into a cusp. Thus, if we decompose according to its visits to the thick region, the collars and the cusps, then the fact that permits to check that it suffices to study the passages of through collars in order to get a lower bound on .

Next, if is a subarc of crossing a collar around a short closed geodesic , then we can apply Dehn twists to to get a family of simple arcs indexed by giving a “contribution” to of

for some constant depending only on the topology of . In this way, the desired result follows by putting all “contributions” together.

### 254A announcement: Analytic prime number theory

In the fall quarter (starting Sep 27) I will be teaching a graduate course on analytic prime number theory. This will be similar to a graduate course I taught in 2015, and in particular will reuse several of the lecture notes from that course, though it will also incorporate some new material (and omit some material covered in the previous course, to compensate). I anticipate covering the following topics:

- Elementary multiplicative number theory
- Complex-analytic multiplicative number theory
- The entropy decrement argument
- Bounds for exponential sums
- Zero density theorems
- Halasz’s theorem and the Matomaki-Radziwill theorem
- The circle method
- (If time permits) Chowla’s conjecture and the Erdos discrepancy problem

Lecture notes for topics 3, 6, and 8 will be forthcoming.

### On the low regularity conjugacy classes of self-similar interval exchange transformations of the Eierlegende Wollmilchsau and Ornithorynque

The celebrated works of several mathematicians (including Poincaré, Denjoy, …, Arnold, Herman, Yoccoz, …) provide a very satisfactory picture of the dynamics of smooth circle diffeomorphisms:

- each -diffeomorphism of the circle has a well-defined rotation number (which can be defined using the cyclic order of its orbits, for instance);
- is
*topologically semi-conjugated*to the rigid rotation (i.e., for a surjective continuous map ) whenever its rotation number is irrational; - if has irrational rotation number , then is
*topologically conjugated*to (i.e., there is an*homeomorphism*such that ); - if , , has an irrational rotation number satisfying a
*Diophantine condition*of the form for some , , and all , then there exists conjugating and (i.e., ); - etc.

In particular, if has *Roth type* (i.e., for all , there exists such that for all ), then any with rotation number is conjugated to whenever . (The nomenclature is motivated by Roth’s theorem saying that any irrational algebraic number has Roth type, and it is well-known that the set of Roth type numbers has full Lebesgue measure in .)

In the last twenty years, many authors gave important contributions towards the extension of this beautiful theory.

In this direction, a particularly successful line of research consists into thinking of circle rotations as standard interval exchange transformations on 2 intervals and trying to build smooth conjugations between generalized interval exchange transformations (g.i.e.t.) and standard interval exchange transformations. In fact, Marmi–Moussa–Yoccoz studied the notion of standard i.e.t. of *restricted Roth type* (a concept designed so that the circle rotation has restricted Roth type [when viewed as an i.e.t. on 2 intervals] if and only if has Roth type) and proved that, for any , the g.i.e.t.s close to a standard i.e.t. of restricted Roth type such that is -conjugated to form a -submanifold of codimension where is the first return map to an interval transverse to a translation flow on a translation surface of genus and is an i.e.t. on intervals.

An interesting consequence of this result of Marmi–Moussa–Yoccoz is the fact that local conjugacy classes behave differently for circle rotations and arbitrary i.e.t.s. Indeed, a circle rotation is an i.e.t. on 2 intervals associated to the first return map of a translation flow on the torus , so that has genus and also . Hence, Marmi–Moussa–Yoccoz theorem says that its local conjugacy class of with of Roth type has codimension *regardless* of the differentiability scale . Of course, this fact was previously known from the theory of circle diffeomorphisms: by the results of Herman and Yoccoz, the sole *obstruction* to obtain a smooth conjugation between and (with of Roth type) is described by a single parameter, namely, the rotation number of . On the other hand, Marmi–Moussa–Yoccoz theorem says that the codimension

of the local conjugacy class of an i.e.t. of restricted Roth type with genus *grows* linearly with the differentiability scale .

**Remark 1** *This indicates that KAM theoretical approaches to the study of the dynamics of g.i.e.t.s might be delicate because the “loss of regularity” in the usual KAM schemes forces the analysis of cohomological equations (linearized versions of the conjugacy problem) in several differentiability scales and Marmi–Moussa–Yoccoz theorem says that these changes of differentiabilty scale produce non-trivial effects on the numbers of obstructions (“codimensions”) to solve cohomological equations.*

In any case, this interesting phenomenon concerning the codimension of local conjugacy classes of i.e.t.s of genus led Marmi–Moussa–Yoccoz to make a series of conjectures (cf. Section 1.2 of their paper) in order to further compare the local conjugacy classes of circle rotations and i.e.t.s of genus .

Among these fascinating conjectures, the second open problem in Section 1.2 of Marmi–Moussa–Yoccoz paper asks whether, for *almost all* i.e.t.s , any g.i.e.t. with trivial conjugacy invariants (e.g., “simple deformations”) and conjugated to is also conjugated to . In other words, the and conjugacy classes of a *typical* i.e.t. coincide.

In this short post, I would like to transcript below some remarks made during recent conversations with Pascal Hubert showing that the hypothesis “for *almost all* i.e.t.s ” can *not* be removed from the conjecture above. In a nutshell, we will see in the sequel that the self-similar standard interval exchange transformations associated to two special translation surfaces (called *Eierlegende Wollmilchsau* and *Ornithorynque*) of genera and are but not conjugated to a rich family of piecewise affine interval exchange transformations. Of course, I think that these examples are probably well-known to experts (and Jean-Christophe Yoccoz was probably aware of them by the time Marmi–Moussa–Yoccoz wrote down their conjectures), but I’m including some details of the construction of these examples here mostly for my own benefit.

**Disclaimer:** As usual, even though the content of this post arose from conversations with Pascal, all mistakes/errors in the sequel are my sole responsibility.

**1. Preliminaries**

**1.1. Rauzy–Veech algorithm**

The notion of “irrational rotation number” for generalized interval exchange transformations relies on the so-called *Rauzy–Veech algorithm*.

More concretely, given a -g.i.e.t. sending a finite partition (modulo zero) of into closed subintervals disposed accordingly to a bijection to a finite partition (modulo zero) of into closed subintervals disposed accordingly to a bijection (via -diffeomorphisms ), an elementary step of the *Rauzy–Veech algorithm* produces a new -g.i.e.t. by taking the first return map of to the interval where , resp. whenever , resp. (and is not defined when ).

We say that a -g.i.e.t. has *irrational rotation number* whenever the Rauzy–Veech algorithm can be iterated indefinitely. This nomenclature is partly justified by the fact that Yoccoz generalized the proof of Poincaré’s theorem in order to establish that a -g.i.e.t. with irrational rotation number is *topologically semi-conjugated* to a standard, minimal i.e.t. .

**1.2. Denjoy counterexamples**

Similarly to Denjoy’s theorem in the case of circle diffeomorphisms, the *obstruction* to promote topological semi-conjugations between and as above into -conjugations is the presence of *wandering intervals* for , i.e., non-trivial intervals whose iterates under are pairwise disjoint (i.e., for all , ).

Moreover, as it was also famously established by Denjoy, a little bit of smoothness (e.g., with derivative of bounded variation) suffices to preclude the existence of wandering intervals for circle diffeomorphisms, and, actually, some smoothness is needed because there are several examples of -diffeomorphisms with any prescribed irrational rotation number and possessing wandering intervals. Nevertheless, it was pointed out by several authors (including Camelier–Gutierrez, Bressaud–Hubert–Maas, Marmi–Moussa–Yoccoz, …), a high amount of smoothness is *not* enough to avoid wandering intervals for arbitrary -g.i.e.t.: indeed, there are *many* examples of piecewise *affine* interval exchange transformations possessing wandering intervals.

**Remark 2** *The facts mentioned in the previous two paragraphs partly justifies the nomenclature Denjoy counterexample for a -g.i.e.t. with irrational rotation number possessing wandering intervals.*

In the context of piecewise affine i.e.t.s, the Denjoy counterexamples are also characterized by the behavior of certain *Birkhoff sums*. More concretely, let be a piecewise affine i.e.t. with irrational rotation number, say is semi-conjugated to a standard i.e.t. . By definition, the logarithm of the slope of is constant on the continuity intervals of and, hence, it allows to naturally define a function taking a constant value on each continuity interval of . In this setting, it is possible to prove (see, e.g., the subsection 3.3.2 of Marmi–Moussa–Yoccoz paper) that has wandering intervals if and only if there exists a point with bi-infinite -orbit such that

where the Birkhoff sum at a point with orbit for all is defined as , resp. for , resp. .

For our subsequent purposes, it is worth to record the following interesting (direct) consequence of this “Birkhoff sums” characterization of piecewise affine Denjoy counterexamples:

**Proposition 1** *Let be a piecewise affine i.e.t. topologically semi-conjugated to a standard, minimal i.e.t. . Denote by the piecewise constant function associated to the logarithms of the slopes of .**If for all with bi-infinite -orbit, then is topologically conjugated to (i.e., is not a Denjoy counterexample).*

**1.3. Special Birkhoff sums and the Kontsevich–Zorich cocycle**

An elementary step of the Rauzy–Veech algorithm replaces a standard, minimal i.e.t. on an interval by a standard, minimal i.e.t. given by the first return map of on an appropriate subinterval .

The *special Birkhoff sum* associated to an elementary step is the operator mapping a function to a function , , where stands for the first return time to .

The special Birkhoff sum operator preserves the space of piecewise constant functions in the sense that is constant on each whenever is constant on each . In particular, the restriction of to the space of such piecewise constant functions gives rise to a matrix . The family of matrices obtained from the successive iterates of the Rauzy–Veech algorithm provides a concrete description of the so-called *Kontsevich–Zorich cocycle*.

In summary, the behaviour of special Birkhoff sums (i.e., Birkhoff sums at certain “return” times) of piecewise constant functions is described by the Kontsevich–Zorich cocycle. Therefore, in view of Proposition 1, it is probably not surprising to the reader at this point that the Lyapunov exponents of the Kontsevich–Zorich cocycle will have something to do with the presence or absence of piecewise affine Denjoy counterexamples.

**1.4. Eierlegende Wollmilchsau and Ornithorynque**

The Eierlegende Wollmilchsau and Ornithorynque are two remarkable translation surfaces and of genera and obtained from finite branched covers of the torus . Among their several curious features, we would like to point out that the following fact proved by Jean-Christophe Yoccoz and myself: if is a standard i.e.t. on or intervals (resp.) associated to the first return map of the translation flow in a typical direction on or (resp.), then there are vectors , and a -dimensional vector subspace such that is an equivariant decomposition with respect to the matrices of the Kontsevich–Zorich cocycle with the following properties:

- (a) generates the Oseledets direction of the top Lyapunov exponent ;
- (b) generates the Oseledets direction of the smallest Lyapunov exponent ;
- (c) the matrices of the Kontsevich–Zorich cocycle act on through a
*finite*group.

In the literature, the Lyapunov exponents are usually called the *tautological* exponents of the Kontsevich–Zorich cocycle. In this terminology, the third item above is saying that all non-tautological Lyapunov exponents of the Kontsevich–Zorich associated to and vanish.

In the next two sections, we will see that this curious behaviour of the Kontsevich–Zorich cocycle of or along allows to construct plenty of piecewise affine i.e.t.s which are but not conjugated to standard (and uniquely ergodic) i.e.t.s.

**2. “Il n’y a pas de contre-exemple de Denjoy affine par morceaux issu de et ”**

In this section (whose title is an obvious reference to a famous article by Jean-Christophe Yoccoz), we will see that the Eierlegende Wollmilchsau and Ornithorynque never produce piecewise affine Denjoy counterexamples with irrational rotation number of “bounded type”.

More precisely, let us consider is a piecewise affine i.e.t. topologically semi-conjugated to coming from (the first return map of the translation flow in the direction of a pseudo-Anosov homeomorphism of) or . It is well-known that the piecewise constant function associated to the logarithms of the slopes of belongs to (see, e.g., Section 3.4 of Marmi–Moussa–Yoccoz paper). In order to simplify the exposition, we assume that the “irrational rotation number” has “bounded type”, that is, is self-similar in the sense that some of its iterates under the Rauzy–Veech algorithm actually coincides with up to scaling.

If , then the item (c) from Subsection 1 above implies that all special Birkhoff sums of (in the future and in the past) are bounded. From this fact, we conclude that for all with bi-infinite -orbit: indeed, as it is explained in details in Bressaud–Bufetov–Hubert article, if is self-similar, then the orbits of can be described by a *substitution* on a finite alphabet and this allows to select a bounded subsequence of thanks to the repetition of certain words in the *prefix-suffix decomposition*.

In particular, it follows from Proposition 1 above that there is *no* Denjoy counterexample among the piecewise affine i.e.t.s topologically semi-conjugated to a self-similar standard i.e.t. coming from or such that .

**Remark 3** *Actually, it is possible to explore the fact that is a stable vector (i.e., it generates the Oseledets space of a negative Lyapunov exponent) to remove the constraint “” from the statement of the previous paragraph.*

In other words, we showed that any *always* provides a piecewise affine i.e.t. -conjugated to . Note that this is a relatively rich family of piecewise affine i.e.t.s because is a vector space of dimension , resp. , when is a self-similar standard i.e.t. coming from , resp. .

**3. Cohomological obstructions to conjugations**

Closing this post, we will show that the elements always lead to piecewise affine i.e.t.s which are *not* conjugated to self-similar standard i.e.t.s of or . Of course, this shows that the and conjugacy classes of a self-similar standard i.e.t. of or are distinct and, *a fortiori*, the Marmi–Moussa–Yoccoz conjecture about the coincidence of and conjugacy classes of standard i.e.t.s becomes false if we remove “for almost all standard i.e.t.s” from its statement.

Suppose that is a piecewise affine i.e.t. -conjugated to a self-similar standard i.e.t. of or , say for some -diffeomorphism . By taking derivatives, we get

since is an isometry. Of course, we recognize the slope of on the left-hand side of the previous equation. So, by taking logarithms, we obtain

where is a function. In other terms, is a solution of the cohomological equation and is a -coboundary. Hence, the Birkhoff sums are bounded and, by continuity of , the special Birkhoff sums of converge to zero. Equivalently, belongs to the *weak stable space* of the Kontsevich–Zorich cocycle (compare with Remark 3.9 of Marmi–Moussa–Yoccoz paper).

However, the item (c) from Subsection 1.4 above tells that the Kontsevich–Zorich cocycle acts on through a finite group of matrices and, thus, can *not* converge to zero under the Kontsevich–Zorich cocycle.

This contradiction proves that is not -conjugated to , as desired.

### Large prime gaps and probabilistic models

William Banks, Kevin Ford, and I have just uploaded to the arXiv our paper “Large prime gaps and probabilistic models“, submitted to Inventiones. In this paper we introduce a random model to help understand the connection between two well known conjectures regarding the primes , the Cramér conjecture and the Hardy-Littlewood conjecture:

**Conjecture 1 (Cramér conjecture)** If is a large number, then the largest prime gap in is of size . (Granville refines this conjecture to , where . Here we use the asymptotic notation for , for , for , and for .)

**Conjecture 2 (Hardy-Littlewood conjecture)** If are fixed distinct integers, then the number of numbers with all prime is as , where the singular series is defined by the formula

(One can view these conjectures as modern versions of two of the classical Landau problems, namely Legendre’s conjecture and the twin prime conjecture respectively.)

A well known connection between the Hardy-Littlewood conjecture and prime gaps was made by Gallagher. Among other things, Gallagher showed that if the Hardy-Littlewood conjecture was true, then the prime gaps with were asymptotically distributed according to an exponential distribution of mean , in the sense that

as for any fixed . Roughly speaking, the way this is established is by using the Hardy-Littlewood conjecture to control the mean values of for fixed , where ranges over the primes in . The relevance of these quantities arises from the Bonferroni inequalities (or “Brun pure sieve“), which can be formulated as the assertion that

when is even and

when is odd, for any natural number ; setting and taking means, one then gets upper and lower bounds for the probability that the interval is free of primes. The most difficult step is to control the mean values of the singular series as ranges over -tuples in a fixed interval such as .

Heuristically, if one extrapolates the asymptotic (1) to the regime , one is then led to Cramér’s conjecture, since the right-hand side of (1) falls below when is significantly larger than . However, this is not a rigorous derivation of Cramér’s conjecture from the Hardy-Littlewood conjecture, since Gallagher’s computations only establish (1) for *fixed* choices of , which is only enough to establish the far weaker bound , which was already known (see this previous paper for a discussion of the best known unconditional lower bounds on ). An inspection of the argument shows that if one wished to extend (1) to parameter choices that were allowed to grow with , then one would need as input a stronger version of the Hardy-Littlewood conjecture in which the length of the tuple , as well as the magnitudes of the shifts , were also allowed to grow with . Our initial objective in this project was then to quantify exactly what strengthening of the Hardy-Littlewood conjecture would be needed to rigorously imply Cramer’s conjecture. The precise results are technical, but roughly we show results of the following form:

**Theorem 3 (Large gaps from Hardy-Littlewood, rough statement)**

- If the Hardy-Littlewood conjecture is uniformly true for -tuples of length , and with shifts of size , with a power savings in the error term, then .
- If the Hardy-Littlewood conjecture is “true on average” for -tuples of length and shifts of size for all , with a power savings in the error term, then .

In particular, we can recover Cramer’s conjecture given a sufficiently powerful version of the Hardy-Littlewood conjecture “on the average”.

Our proof of this theorem proceeds more or less along the same lines as Gallagher’s calculation, but now with allowed to grow slowly with . Again, the main difficulty is to accurately estimate average values of the singular series . Here we found it useful to switch to a probabilistic interpretation of this series. For technical reasons it is convenient to work with a truncated, unnormalised version

of the singular series, for a suitable cutoff ; it turns out that when studying prime tuples of size , the most convenient cutoff is the “Pólya magic cutoff“, defined as the largest prime for which

(this is well defined for ); by Mertens’ theorem, we have . One can interpret probabilistically as

where is the randomly sifted set of integers formed by removing one residue class uniformly at random for each prime . The Hardy-Littlewood conjecture can be viewed as an assertion that the primes behave in some approximate statistical sense like the random sifted set , and one can prove the above theorem by using the Bonferroni inequalities both for the primes and for the random sifted set, and comparing the two (using an even for the sifted set and an odd for the primes in order to be able to combine the two together to get a useful bound).

The proof of Theorem 3 ended up not using any properties of the set of primes other than that this set obeyed some form of the Hardy-Littlewood conjectures; the theorem remains true (with suitable notational changes) if this set were replaced by any other set. In order to convince ourselves that our theorem was not vacuous due to our version of the Hardy-Littlewood conjecture being too strong to be true, we then started exploring the question of coming up with random models of which obeyed various versions of the Hardy-Littlewood and Cramér conjectures.

This line of inquiry was started by Cramér, who introduced what we now call the *Cramér random model* of the primes, in which each natural number is selected for membership in with an independent probability of . This model matches the primes well in some respects; for instance, it almost surely obeys the “Riemann hypothesis”

and Cramér also showed that the largest gap was almost surely . On the other hand, it does not obey the Hardy-Littlewood conjecture; more precisely, it obeys a simplified variant of that conjecture in which the singular series is absent.

Granville proposed a refinement to Cramér’s random model in which one first sieves out (in each dyadic interval ) all residue classes for for a certain threshold , and then places each surviving natural number in with an independent probability . One can verify that this model obeys the Hardy-Littlewood conjectures, and Granville showed that the largest gap in this model was almost surely , leading to his conjecture that this bound also was true for the primes. (Interestingly, this conjecture is not yet borne out by numerics; calculations of prime gaps up to , for instance, have shown that never exceeds in this range. This is not necessarily a conflict, however; Granville’s analysis relies on inspecting gaps in an extremely sparse region of natural numbers that are more devoid of primes than average, and this region is not well explored by existing numerics. See this previous blog post for more discussion of Granville’s argument.)

However, Granville’s model does not produce a power savings in the error term of the Hardy-Littlewood conjectures, mostly due to the need to truncate the singular series at the logarithmic cutoff . After some experimentation, we were able to produce a tractable random model for the primes which obeyed the Hardy-Littlewood conjectures with power savings, and which reproduced Granville’s gap prediction of (we also get an upper bound of for both models, though we expect the lower bound to be closer to the truth); to us, this strengthens the case for Granville’s version of Cramér’s conjecture. The model can be described as follows. We select one residue class uniformly at random for each prime , and as before we let be the sifted set of integers formed by deleting the residue classes with . We then set

with Pólya’s magic cutoff (this is the cutoff that gives a density consistent with the prime number theorem or the Riemann hypothesis). As stated above, we are able to show that almost surely one has

and that the Hardy-Littlewood conjectures hold with power savings for up to for any fixed and for shifts of size . This is unfortunately a tiny bit weaker than what Theorem 3 requires (which more or less corresponds to the endpoint ), although there is a variant of Theorem 3 that can use this input to produce a lower bound on gaps in the model (but it is weaker than the one in (3)). In fact we prove a more precise almost sure asymptotic formula for that involves the optimal bounds for the *linear sieve* (or *interval sieve*), in which one deletes one residue class modulo from an interval for all primes up to a given threshold. The lower bound in (3) relates to the case of deleting the residue classes from ; the upper bound comes from the delicate analysis of the linear sieve by Iwaniec. Improving on either of the two bounds looks to be quite a difficult problem.

The probabilistic analysis of is somewhat more complicated than of or as there is now non-trivial coupling between the events as varies, although moment methods such as the second moment method are still viable and allow one to verify the Hardy-Littlewood conjectures by a lengthy but fairly straightforward calculation. To analyse large gaps, one has to understand the statistical behaviour of a random linear sieve in which one starts with an interval and randomly deletes a residue class for each prime up to a given threshold. For very small this is handled by the deterministic theory of the linear sieve as discussed above. For medium sized , it turns out that there is good concentration of measure thanks to tools such as Bennett’s inequality or Azuma’s inequality, as one can view the sieving process as a martingale or (approximately) as a sum of independent random variables. For larger primes , in which only a small number of survivors are expected to be sieved out by each residue class, a direct combinatorial calculation of all possible outcomes (involving the random graph that connects interval elements to primes if falls in the random residue class ) turns out to give the best results.

### Quantitative bounds for critically bounded solutions to the Navier-Stokes equations

I’ve just uploaded to the arXiv my paper “Quantitative bounds for critically bounded solutions to the Navier-Stokes equations“, submitted to the proceedings of the Linde Hall Inaugural Math Symposium. (I unfortunately had to cancel my physical attendance at this symposium for personal reasons, but was still able to contribute to the proceedings.) In recent years I have been interested in working towards establishing the existence of classical solutions for the Navier-Stokes equations

that blow up in finite time, but this time for a change I took a look at the other side of the theory, namely the conditional regularity results for this equation. There are several such results that assert that if a certain norm of the solution stays bounded (or grows at a controlled rate), then the solution stays regular; taken in the contrapositive, they assert that if a solution blows up at a certain finite time , then certain norms of the solution must also go to infinity. Here are some examples (not an exhaustive list) of such blowup criteria:

- (Leray blowup criterion, 1934) If blows up at a finite time , and , then for an absolute constant .
- (Prodi–Serrin–Ladyzhenskaya blowup criterion, 1959-1967) If blows up at a finite time , and , then , where .
- (Beale-Kato-Majda blowup criterion, 1984) If blows up at a finite time , then , where is the vorticity.
- (Kato blowup criterion, 1984) If blows up at a finite time , then for some absolute constant .
- (Escauriaza-Seregin-Sverak blowup criterion, 2003) If blows up at a finite time , then .
- (Seregin blowup criterion, 2012) If blows up at a finite time , then .
- (Phuc blowup criterion, 2015) If blows up at a finite time , then for any .
- (Gallagher-Koch-Planchon blowup criterion, 2016) If blows up at a finite time , then for any .
- (Albritton blowup criterion, 2016) If blows up at a finite time , then for any .

My current paper is most closely related to the Escauriaza-Seregin-Sverak blowup criterion, which was the first to show a critical (i.e., scale-invariant, or dimensionless) spatial norm, namely , had to become large. This result now has many proofs; for instance, many of the subsequent blowup criterion results imply the Escauriaza-Seregin-Sverak one as a special case, and there are also additional proofs by Gallagher-Koch-Planchon (building on ideas of Kenig-Koch), and by Dong-Du. However, all of these proofs rely on some form of a compactness argument: given a finite time blowup, one extracts some suitable family of rescaled solutions that converges in some weak sense to a limiting solution that has some additional good properties (such as almost periodicity modulo symmetries), which one can then rule out using additional qualitative tools, such as unique continuation and backwards uniqueness theorems for parabolic heat equations. In particular, all known proofs use some version of the backwards uniqueness theorem of Escauriaza, Seregin, and Sverak. Because of this reliance on compactness, the existing proofs of the Escauriaza-Seregin-Sverak blowup criterion are qualitative, in that they do not provide any quantitative information on how fast the norm will go to infinity (along a subsequence of times).

On the other hand, it is a general principle that qualitative arguments established using compactness methods ought to have quantitative analogues that replace the use of compactness by more complicated substitutes that give effective bounds; see for instance these previous blog posts for more discussion. I therefore was interested in trying to obtain a quantitative version of this blowup criterion that gave reasonably good effective bounds (in particular, my objective was to avoid truly enormous bounds such as tower-exponential or Ackermann function bounds, which often arise if one “naively” tries to make a compactness argument effective). In particular, I obtained the following triple-exponential quantitative regularity bounds:

**Theorem 1** If is a classical solution to Navier-Stokes on with

and

for and .

As a corollary, one can now improve the Escauriaza-Seregin-Sverak blowup criterion to

for some absolute constant , which to my knowledge is the first (*very* slightly) supercritical blowup criterion for Navier-Stokes in the literature.

The proof uses many of the same quantitative inputs as previous arguments, most notably the Carleman inequalities used to establish unique continuation and backwards uniqueness theorems for backwards heat equations, but also some additional techniques that make the quantitative bounds more efficient. The proof focuses initially on points of concentration of the solution, which we define as points where there is a frequency for which one has the bound

for a large absolute constant , where is a Littlewood-Paley projection to frequencies . (This can be compared with the upper bound of for the quantity on the left-hand side that follows from (1).) The factor of normalises the left-hand side of (2) to be dimensionless (i.e., critical). The main task is to show that the dimensionless quantity cannot get too large; in particular, we end up establishing a bound of the form

from which the above theorem ends up following from a routine adaptation of the local well-posedness and regularity theory for Navier-Stokes.

The strategy is to show that any concentration such as (2) when is large must force a significant component of the norm of to also show up at many other locations than , which eventually contradicts (1) if one can produce enough such regions of non-trivial norm. (This can be viewed as a quantitative variant of the “rigidity” theorems in some of the previous proofs of the Escauriaza-Seregin-Sverak theorem that rule out solutions that exhibit too much “compactness” or “almost periodicity” in the topology.) The chain of causality that leads from a concentration (2) at to significant norm at other regions of the time slice is somewhat involved (though simpler than the much more convoluted schemes I initially envisaged for this argument):

- Firstly, by using Duhamel’s formula, one can show that a concentration (2) can only occur (with large) if there was also a preceding concentration
at some slightly previous point in spacetime, with also close to (more precisely, we have , , and ). This can be viewed as a sort of contrapositive of a “local regularity theorem”, such as the ones established by Caffarelli, Kohn, and Nirenberg. A key point here is that the lower bound in the conclusion (3) is precisely the same as the lower bound in (2), so that this backwards propagation of concentration can be iterated.

- Iterating the previous step, one can find a sequence of concentration points
with the propagating backwards in time; by using estimates ultimately resulting from the dissipative term in the energy identity, one can extract such a sequence in which the increase geometrically with time, the are comparable (up to polynomial factors in ) to the natural frequency scale , and one has . Using the “epochs of regularity” theory that ultimately dates back to Leray, and tweaking the slightly, one can also place the times in intervals (of length comparable to a small multiple of ) in which the solution is quite regular (in particular, enjoy good bounds on ).

- The concentration (4) can be used to establish a lower bound for the norm of the vorticity near . As is well known, the vorticity obeys the vorticity equation
In the epoch of regularity , the coefficients of this equation obey good bounds, allowing the machinery of Carleman estimates to come into play. Using a Carleman estimate that is used to establish unique continuation results for backwards heat equations, one can propagate this lower bound to also give lower bounds on the vorticity (and its first derivative) in annuli of the form for various radii , although the lower bounds decay at a gaussian rate with .

- Meanwhile, using an energy pigeonholing argument of Bourgain (which, in this Navier-Stokes context, is actually an enstrophy pigeonholing argument), one can locate some annuli where (a slightly normalised form of) the entrosphy is small at time ; using a version of the localised enstrophy estimates from a previous paper of mine, one can then propagate this sort of control forward in time, obtaining an “annulus of regularity” of the form in which one has good estimates; in particular, one has type bounds on in this cylindrical annulus.
- By intersecting the previous epoch of regularity with the above annulus of regularity, we have some lower bounds on the norm of the vorticity (and its first derivative) in the annulus of regularity. Using a Carleman estimate first introduced by Escauriaza, Seregin, and Sverak, as well as a second application of the Carleman estimate used previously, one can then propagate this lower bound back up to time , establishing a lower bound for the vorticity on the spatial annulus . By some basic Littlewood-Paley theory one can parlay this lower bound to a lower bound on the norm of the velocity ; crucially, this lower bound is uniform in .
- If is very large (triple exponential in !), one can then find enough scales with disjoint annuli that the total lower bound on the norm of provided by the above arguments is inconsistent with (1), thus establishing the claim.

The chain of causality is summarised in the following image:

It seems natural to conjecture that similar triply logarithmic improvements can be made to several of the other blowup criteria listed above, but I have not attempted to pursue this question. It seems difficult to improve the triple logarithmic factor using only the techniques here; the Bourgain pigeonholing argument inevitably costs one exponential, the Carleman inequalities cost a second, and the stacking of scales at the end to contradict the upper bound costs the third.

### Eigenvectors from eigenvalues

Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv the short unpublished note “Eigenvectors from eigenvalues“. This note gives two proofs of a general eigenvector identity observed recently by Denton, Parke and Zhang in the course of some quantum mechanical calculations. The identity is as follows:

**Theorem 1** Let be an Hermitian matrix, with eigenvalues . Let be a unit eigenvector corresponding to the eigenvalue , and let be the component of . Then

where is the Hermitian matrix formed by deleting the row and column from .

For instance, if we have

for some real number , -dimensional vector , and Hermitian matrix , then we have

assuming that the denominator is non-zero.

Once one is aware of the identity, it is not so difficult to prove it; we give two proofs, each about half a page long, one of which is based on a variant of the Cauchy-Binet formula, and the other based on properties of the adjugate matrix. But perhaps it is surprising that such a formula exists at all; one does not normally expect to learn much information about eigenvectors purely from knowledge of eigenvalues. In the random matrix theory literature, for instance in this paper of Erdos, Schlein, and Yau, or this later paper of Van Vu and myself, a related identity has been used, namely

but it is not immediately obvious that one can derive the former identity from the latter. (I do so below the fold; we ended up not putting this proof in the note as it was longer than the two other proofs we found. I also give two other proofs below the fold, one from a more geometric perspective and one proceeding via Cramer’s rule.) It was certainly something of a surprise to me that there is no explicit appearance of the components of in the formula (1) (though they do indirectly appear through their effect on the eigenvalues ; for instance from taking traces one sees that ).

One can get some feeling of the identity (1) by considering some special cases. Suppose for instance that is a diagonal matrix with all distinct entries. The upper left entry of is one of the eigenvalues of . If it is equal to , then the eigenvalues of are the other eigenvalues of , and now the left and right-hand sides of (1) are equal to . At the other extreme, if is equal to a different eigenvalue of , then now appears as an eigenvalue of , and both sides of (1) now vanish. More generally, if we order the eigenvalues and , then the Cauchy interlacing inequalities tell us that

for , and

for , so that the right-hand side of (1) lies between and , which is of course consistent with (1) as is a unit vector. Thus the identity relates the coefficient sizes of an eigenvector with the extent to which the Cauchy interlacing inequalities are sharp.

** — 1. Relating the two identities — **

We now show how (1) can be deduced from (2). By a limiting argument, it suffices to prove (1) in the case when is not an eigenvalue of . Without loss of generality we may take . By subtracting the matrix from (and from , thus shifting all the eigenvalues down by , we may also assume without loss of generality that . So now we wish to show that

The right-hand side is just . If one differentiates the characteristic polynomial

at , one sees that

Finally, (2) can be rewritten as

so our task is now to show that

By Schur complement, we have

Since is an eigenvalue of , but not of (by hypothesis), the factor vanishes when . If we then differentiate (4) in and set we obtain (3) as desired.

** — 2. A geometric proof — **

Here is a more geometric way to think about the identity. One can view as a linear operator on (mapping to for any vector ); it then also acts on all the exterior powers by mapping to for all vectors . In particular, if one evaluates on the basis of induced by the orthogonal eigenbasis , we see that the action of on is rank one, with

for any , where is the inner product on induced by the standard inner product on . If we now apply this to the -form , we have , while is equal to plus some terms orthogonal to . Since , Theorem 1 follows.

** — 3. A proof using Cramer’s rule — **

By a limiting argument we can assume that all the eigenvalues of are simple. From the spectral theorem we can compute the resolvent for as

Extracting the component of both sides and using Cramer’s rule, we conclude that

or in terms of eigenvalues

Both sides are rational functions with a simple pole at the eigenvalues . Extracting the residue at we conclude that

and Theorem 1 follows. (Note that this approach also gives a formula for for , although the formula becomes messier when because the relevant minor of is no longer a scalar multiple of the identity .)

### Sharp bounds for multilinear curved Kakeya, restriction and oscillatory integral estimates away from the endpoint

I have just uploaded to the arXiv my paper “Sharp bounds for multilinear curved Kakeya, restriction and oscillatory integral estimates away from the endpoint“, submitted to Mathematika. In this paper I return (after more than a decade’s absence) to one of my first research interests, namely the Kakeya and restriction family of conjectures. The starting point is the following “multilinear Kakeya estimate” first established in the non-endpoint case by Bennett, Carbery, and myself, and then in the endpoint case by Guth (with further proofs and extensions by Bourgain-Guth and Carbery-Valdimarsson:

**Theorem 1 (Multilinear Kakeya estimate)** Let be a radius. For each , let denote a finite family of infinite tubes in of radius . Assume the following axiom:

- (i) (Transversality) whenever is oriented in the direction of a unit vector for , we have
for some , where we use the usual Euclidean norm on the wedge product .

where are the usual Lebesgue norms with respect to Lebesgue measure, denotes the indicator function of , and denotes the cardinality of .

The original proof of this proceeded using a heat flow monotonicity method, which in my previous post I reinterpreted using a “virtual integration” concept on a fractional Cartesian product space. It turns out that this machinery is somewhat flexible, and can be used to establish some other estimates of this type. The first result of this paper is to extend the above theorem to the curved setting, in which one localises to a ball of radius (and sets to be small), but allows the tubes to be curved in a fashion. If one runs the heat flow monotonicity argument, one now picks up some additional error terms arising from the curvature, but as the spatial scale approaches zero, the tubes become increasingly linear, and as such the error terms end up being an integrable multiple of the main term, at which point one can conclude by Gronwall’s inequality (actually for technical reasons we use a bootstrap argument instead of Gronwall). A key point in this approach is that one obtains optimal bounds (not losing factors of or ), so long as one stays away from the endpoint case (which does not seem to be easily treatable by the heat flow methods). Previously, the paper of Bennett, Carbery, and myself was able to use an induction on scale argument to obtain a curved multilinear Kakeya estimate losing a factor of (after optimising the argument); later arguments of Bourgain-Guth and Carbery-Valdimarsson, based on algebraic topology methods, could also obtain a curved multilinear Kakeya estimate without such losses, but only in the algebraic case when the tubes were neighbourhoods of algebraic curves of bounded degree.

Perhaps more interestingly, we are also able to extend the heat flow monotonicity method to apply directly to the multilinear restriction problem, giving the following global multilinear restriction estimate:

**Theorem 2 (Multilinear restriction theorem)** Let be an exponent, and let be a parameter. Let be a sufficiently large natural number, depending only on . For , let be an open subset of , and let be a smooth function obeying the following axioms:

for any , , extended by zero outside of , and denotes the extension operator

Local versions of such estimate, in which is replaced with for some , and one accepts a loss of the form , were already established by Bennett, Carbery, and myself using an induction on scale argument. In a later paper of Bourgain-Guth these losses were removed by “epsilon removal lemmas” to recover Theorme 2, but only in the case when all the hypersurfaces involved had curvatures bounded away from zero.

There are two main new ingredients in the proof of Theorem 2. The first is to replace the usual induction on scales scheme to establish multilinear restriction by a “ball inflation” induction on scales scheme that more closely resembles the proof of decoupling theorems. In particular, we actually prove the more general family of estimates

where denotes the local energies

(actually for technical reasons it is more convenient to use a smoother weight than the strict cutoff to the disk ). With logarithmic losses, it is not difficult to establish this estimate by an upward induction on . To avoid such losses we use the heat flow monotonicity method. Here we run into the issue that the extension operators are complex-valued rather than non-negative, and thus would not be expected to obey many good montonicity properties. However, the local energies can be expressed in terms of the magnitude squared of what is essentially the Gabor transform of , and these are non-negative; furthermore, the dispersion relation associated to the extension operators implies that these Gabor transforms propagate along tubes, so that the situation becomes quite similar (up to several additional lower order error terms) to that in the multilinear Kakeya problem. (This can be viewed as a continuous version of the usual wave packet decomposition method used to relate restriction and Kakeya problems, which when combined with the heat flow monotonicity method allows for one to use a continuous version of induction on scales methods that do not concede any logarithmic factors.)

Finally, one can combine the curved multilinear Kakeya result with the multilinear restriction result to obtain estimates for multilinear oscillatory integrals away from the endpoint. Again, this sort of implication was already established in the previous paper of Bennett, Carbery, and myself, but the arguments there had some epsilon losses in the exponents; here we were able to run the argument more carefully and avoid these losses.

### Twisted convolution and the sensitivity conjecture

Earlier this month, Hao Huang (who, incidentally, was a graduate student here at UCLA) gave a remarkably short proof of a long-standing problem in theoretical computer science known as the sensitivity conjecture. See for instance this blog post of Gil Kalai for further discussion and links to many other online discussions of this result. One formulation of the theorem proved is as follows. Define the -dimensional hypercube graph to be the graph with vertex set , and with every vertex joined to the vertices , where is the standard basis of .

**Theorem 1 (Lower bound on maximum degree of induced subgraphs of hypercube)** Let be a set of at least vertices in . Then there is a vertex in that is adjacent (in ) to at least other vertices in .

The bound (or more precisely, ) is completely sharp, as shown by Chung, Furedi, Graham, and Seymour; we describe this example below the fold. When combined with earlier reductions of Gotsman-Linial and Nisan-Szegedy; we give these below the fold also.

Let be the adjacency matrix of (where we index the rows and columns directly by the vertices in , rather than selecting some enumeration ), thus when for some , and otherwise. The above theorem then asserts that if is a set of at least vertices, then the minor of has a row (or column) that contains at least non-zero entries.

The key step to prove this theorem is the construction of rather curious variant of the adjacency matrix :

**Proposition 2** There exists a matrix which is entrywise dominated by in the sense that

and such that has as an eigenvalue with multiplicity .

Assuming this proposition, the proof of Theorem 1 can now be quickly concluded. If we view as a linear operator on the -dimensional space of functions of , then by hypothesis this space has a -dimensional subspace on which acts by multiplication by . If is a set of at least vertices in , then the space of functions on has codimension at most in , and hence intersects non-trivially. Thus the minor of also has as an eigenvalue (this can also be derived from the Cauchy interlacing inequalities), and in particular this minor has operator norm at least . By Schur’s test, this implies that one of the rows or columns of this matrix has absolute values summing to at least , giving the claim.

**Remark 3** The argument actually gives a strengthening of Theorem 1: there exists a vertex of with the property that for every natural number , there are at least paths of length in the restriction of to that start from . Indeed, if we let be an eigenfunction of on , and let be a vertex in that maximises the value of , then for any we have that the component of is equal to ; on the other hand, by the triangle inequality, this component is at most times the number of length paths in starting from , giving the claim.

This argument can be viewed as an instance of a more general “interlacing method” to try to control the behaviour of a graph on all large subsets by first generating a matrix on with very good spectral properties, which are then partially inherited by the minor of by interlacing inequalities. In previous literature using this method (see e.g., this survey of Haemers, or this paper of Wilson), either the original adjacency matrix , or some non-negatively weighted version of that matrix, was used as the controlling matrix ; the novelty here is the use of signed controlling matrices. It will be interesting to see what further variants and applications of this method emerge in the near future. (Thanks to Anurag Bishoi in the comments for these references.)

The “magic” step in the above argument is constructing . In Huang’s paper, is constructed recursively in the dimension in a rather simple but mysterious fashion. Very recently, Roman Karasev gave an interpretation of this matrix in terms of the exterior algebra on . In this post I would like to give an alternate interpretation in terms of the operation of *twisted convolution*, which originated in the theory of the Heisenberg group in quantum mechanics.

Firstly note that the original adjacency matrix , when viewed as a linear operator on , is a convolution operator

where

is the counting measure on the standard basis , and denotes the ordinary convolution operation

As is well known, this operation is commutative and associative. Thus for instance the square of the adjacency operator is also a convolution operator

where the convolution kernel is moderately complicated:

The factor in this expansion comes from combining the two terms and , which both evaluate to .

More generally, given any bilinear form , one can define the *twisted convolution*

of two functions . This operation is no longer commutative (unless is symmetric). However, it remains associative; indeed, one can easily compute that

In particular, if we define the twisted convolution operator

then the square is also a twisted convolution operator

and the twisted convolution kernel can be computed as

For general bilinear forms , this twisted convolution is just as messy as is. But if we take the specific bilinear form

then for and for , and the above twisted convolution simplifies to

and now is very simple:

Thus the only eigenvalues of are and . The matrix is entrywise dominated by in the sense of (1), and in particular has trace zero; thus the and eigenvalues must occur with equal multiplicity, so in particular the eigenvalue occurs with multiplicity since the matrix has dimensions . This establishes Proposition 2.

**Remark 4** Twisted convolution is actually just a component of ordinary convolution, but not on the original group ; instead it relates to convolution on a Heisenberg group extension of this group. More specifically, define the Heisenberg group to be the set of pairs with group law

and inverse operation

(one can dispense with the negative signs here if desired, since we are in characteristic two). Convolution on is defined in the usual manner: one has

for any . Now if is a function on the original group , we can define the lift by the formula

and then by chasing all the definitions one soon verifies that

for any , thus relating twisted convolution to Heisenberg group convolution .

**Remark 5** With the twisting by the specific bilinear form given by (2), convolution by and now anticommute rather than commute. This makes the twisted convolution algebra isomorphic to a Clifford algebra (the real or complex algebra generated by formal generators subject to the relations for ) rather than the commutative algebra more familiar to abelian Fourier analysis. This connection to Clifford algebra (also observed independently by Tom Mrowka and by Daniel Matthews) may be linked to the exterior algebra interpretation of the argument in the recent preprint of Karasev mentioned above.

**Remark 6** One could replace the form (2) in this argument by any other bilinear form that obeyed the relations and for . However, this additional level of generality does not add much; any such will differ from by an antisymmetric form (so that for all , which in characteristic two implied that for all ), and such forms can always be decomposed as , where . As such, the matrices and are conjugate, with the conjugation operator being the diagonal matrix with entries at each vertex .

**Remark 7** (Added later) This remark combines the two previous remarks. One can view any of the matrices in Remark 6 as components of a single canonical matrix that is still of dimensions , but takes values in the Clifford algebra from Remark 5; with this “universal algebra” perspective, one no longer needs to make any arbitrary choices of form . More precisely, let denote the vector space of functions from the hypercube to the Clifford algebra; as a real vector space, this is a dimensional space, isomorphic to the direct sum of copies of , as the Clifford algebra is itself dimensional. One can then define a canonical Clifford adjacency operator on this space by

where are the generators of . This operator can either be identified with a Clifford-valued matrix or as a real-valued matrix. In either case one still has the key algebraic relations and , ensuring that when viewed as a real matrix, half of the eigenvalues are equal to and half equal to . One can then use this matrix in place of any of the to establish Theorem 1 (noting that Schur’s test continues to work for Clifford-valued matrices because of the norm structure on ).

To relate to the real matrices , first observe that each point in the hypercube can be associated with a one-dimensional real subspace (i.e., a line) in the Clifford algebra by the formula

for any (note that this definition is well-defined even if the are out of order or contain repetitions). This can be viewed as a discrete line bundle over the hypercube. Since for any , we see that the -dimensional real linear subspace of of sections of this bundle, that is to say the space of functions such that for all , is an invariant subspace of . (Indeed, using the left-action of the Clifford algebra on , which commutes with , one can naturally identify with , with the left action of acting purely on the first factor and acting purely on the second factor.) Any trivialisation of this line bundle lets us interpret the restriction of to as a real matrix. In particular, given one of the bilinear forms from Remark 6, we can identify with by identifying any real function with the lift defined by

whenever . A somewhat tedious computation using the properties of then eventually gives the intertwining identity

and so is conjugate to .

** — 1. The Chung, Furedi, Graham, and Seymour example — **

The paper of by Chung, Furedi, Graham, and Seymour gives, for any , an example of a subset of of cardinality for which the maximum degree of restricted to is at most , thus showing that Theorem 1 cannot be improved (beyond the trivial improvement of upgrading to , because the maximum degree is obviously a natural number).

Define the “Möbius function” to be the function

for . This function is extremely balanced on coordinate spaces. Indeed, from the binomial theorem (which uses the convention ) we have

More generally, given any index set of cardinality , we have

Now let be a partition of into disjoint non-empty sets. For each , let be the subspace of consisting of those such that for all . Then for any , we have

and the right-hand side vanishes if and equals when . Applying the inclusion-exclusion principle, we conclude that

and thus also (assuming )

so that

Thus, if denotes the set of those with , together with those with , then has to have two more elements than its complement , and hence has cardinality .

Now observe that, if with and , then , and if then unless . Thus in this case the total number of for which is at most . Conversely, if with and , then , and for each there is at most one that will make lie in . Hence in this case the total number of for which is at most . Thus the maximum degree of the subgraph of induced by is at most . By choosing the to be a partition of into pieces, each of cardinality at most , we obtain the claim.

**Remark 8** Suppose that is a perfect square, then the lower bound here exactly matches the upper bound in Theorem 1. In particular, the minor of the matrix must have an eigenvector of eigenvalue . Such an eigenvector can be explicitly constructed as follows. Let be the vector defined by setting

for some , , , and for all other (one easily verifies that the previous types of lie in ). We claim that

for all . Expanding out the left-hand side, we wish to show that

First suppose that is of the form (3). One checks that lies in precisely when for one of the , in which case

Since , this simplifies using (3) as

giving (5) in this case. Similarly, if is of the form (4), then lies in precisely when , in which case one can argue as before to show that

and (5) again follows. Finally, if is not of either of the two forms (3), (4), one can check that is never of these forms either, and so both sides of (5) vanish.

The same analysis works for any of the other bilinear forms in Remark 6. Using the Clifford-valued operator from Remark 7, the eigenfunction is cleaner; it is defined by

when is of the form (3), and

when is of the form (4), with otherwise.

** — 2. From induced subgraph bounds to the sensitivity conjecture — **

On the hypercube , let denote the functions

The monomials in are then the characters of , so by Fourier expansion every function can be viewed as a polynomial in the (with each monomial containing at most one copy of ; higher powers of each are unnecessary since . In particular, one can meaningfully talk about the degree of a function . Observe also that the Möbius function from the preceding section is just the monomial .

Define the *sensitivity* of a Boolean function to be the largest number for which there is an such that there are at least values of with . Using an argument of Gotsman and Linial, we can now relate the sensitivity of a function to its degree:

**Corollary 9 (Lower bound on sensitivity)** For any boolean function , one has .

*Proof:* Write . By permuting the indices, we may assume that contains a non-trivial multiple of the monomial . By restricting to the subspace (which cannot increase the sensitivity), we may then assume without loss of generality that . The Fourier coefficient of is just the mean value

of times the Möbius function , so this mean value is non-zero. This means that one of the sets or has cardinality at least . Let denote the larger of these two sets. By Theorem 1, there is an such that for at least values of ; since , this implies that for at least values of , giving the claim.

The construction of Chung, Furedi, Graham, and Seymour from the previous section can be easily adapted to show that this lower bound is tight (other than the trivial improvement of replacing by ).

Now we need to digress on some bounds involving polynomials of one variable. We begin with an inequality of Bernstein concerning trigonometric polynomials:

**Lemma 10 (Bernstein inequality)** Let be a trigonometric polynomial of degree at most , that is to say a complex linear combination of for . Then

Observe that equality holds when or . Specialising to linear combinations of , we obtain the classical Bernstein inequality

for complex polynomials of degree at most .

*Proof:* If one is willing to lose a constant factor in this estimate, this bound can be easily established from modern Littlewood-Paley theory (see e.g., Exercise 52 of these lecture notes). Here we use an interlacing argument due to Boas. We first restrict to the case when has real coefficients. We may normalise . Let be a real parameter in . The trigonometric polynomial alternately takes the values and at the values . Thus the trigonometric polynomial alternates in sign at these values, and thus by the intermediate value theorem has a zero on each of the intervals . On the other hand, a trigonometric polynomial of degree at most can be expressed by de Moivre’s theorem as times a complex polynomial in of degree at most , and thus has at most zeroes. Thus we see that has exactly one zero in each . Furthermore, at this zero, the derivative of this function must be positive if is increasing on this interval, and negative if is decreasing on this interval. In summary, we have shown that if and are such that , then has the same sign as . By translating the function , we also conclude that if and are such that for some , then has the same sign as .

If , then we can find such that and is positive, and we conclude that ; thus we have the upper bound

A similar argument (with now chosen to be negative) similarly bounds . This gives the claim for real-valued trigonometric polynomials . (Indeed, this argument even gives the slightly sharper bound .)

To amplify this to complex valued polynomials, we take advantage of phase rotation invariance. If is a complex trigonometric polynomial, then by applying Bernstein’s inequality to the real part we have

But then we can multiply by any complex phase and conclude that

Taking suprema in , one obtains the claim for complex polynomials .

The analogue of Bernstein’s inequality for the unit interval is known as Markov’s inequality for polynomials:

**Lemma 11 (Markov’s inequality for polynomials)** Let be a polynomial of degree . Then

This bound is sharp, as is seen by inspecting the Chebyshev polynomial , defined as the unique polynomial giving the trigonometric identity

Differentiating (6) using the chain rule, we see that

the right-hand side approaches as , demonstrating that the factor here is sharp.

*Proof:* We again use an argument of Boas. We may normalise so that

The function is a trigonometric polynomial of degree at most , so by Bernstein’s inequality and the chain rule we have

for all . This already gives Markov’s inequality except in the edge regions (since ). By reflection symmetry, it then suffices to verify Markov’s inequality in the region .

From (6), the Chebyshev polynomial attains the values alternately at the different points . Thus, if , the polynomial changes sign at least times on , and thus must have all zeroes inside this interval by the intermediate value theorem; furthermore, of these zeroes will lie to the left of . By Rolle’s theorem, the derivative then has all zeroes in the interval , and at least of these will lie to the left of . In particular, the derivative can have at most one zero to once to the right of .

Since , is positive at , and hence positive as since there are no zeroes outside of . Thus the leading coefficient of is positive, which implies the same for its derivative . Thus is positive when .

From (9) one has , hence by (7) we see that is also positive at . Thus cannot become negative for , as this would create at least two zeroes to the right of . We conclude that in this region we have

From (7) we have , and the claim follows.

**Remark 12** The following slightly shorter argument gives the slightly weaker bound . We again use the normalisation (8). By two applications of Bernstein’s inequality, the function has first derivative bounded in magnitude by , and second derivative bounded in magnitude by . As this function also has vanishing first derivative at , we conclude the bounds

and thus by the chain rule

For , one easily checks that the right-hand side is at most , giving the claim.

This implies a result of Ehlich-Zeller and of Rivlin-Cheney:

**Corollary 13 (Discretised Markov inequality)** Let be a polynomial of degree . If

*Proof:* We use an argument of Nisan and Szegedy. Assume for sake of contradiction that , so in particular . From the fundamental theorem of calculus and the triangle inequality one has

By a rescaled and translated version of Markov’s inequality we have

which when inserted into the preceding inequality gives after some rearranging

and then after a second application of (11) gives

Comparing with (10), we conclude that

and the claim follows after some rearranging.

Nisan and Szegedy observed that this one-dimensional degree bound can be lifted to the hypercube by a symmetrisation argument:

**Corollary 14 (Multidimensional Markov inequality bound)** Let be such that and for . Then

*Proof:* By averaging over all permutations of the indices (which can decrease the degree of , but not increase it), we may assume that is a symmetric function of the inputs . Using the Newton identities, we can then write

for some real polynomial of degree at most , where

is the Hamming length of . By hypothesis, , , and , hence by the mean-value theorem . Applying Corollary 13 with , we obtain the claim.

Define the *block sensitivity* of a Boolean function to be the largest number for which there is an such that there are at least disjoint subsets of with for . We have

**Theorem 15 (Sensitivity conjecture)** One has

More precisely, the sensitivity conjecture of Nisan and Szegedy asserted the bound ; Huang’s result thus gives explicit values for the exponents. It is still open whether the exponent in this theorem can be improved; it is known that one cannot improve it to below , by analysing a variant of the Chung-Furedi-Graham-Seymour example (see these notes of Regan for details). *Proof:* The lower bound for is immediate from the definitions, since the sensitivity arises by restricting the in the definition of block sensitivity to singleton sets. To prove the upper bound, it suffices from Proposition 9 to establish the bound

Let . By hypothesis, there are and disjoint subsets of such that for . We may normalise , and . If we then define the pullback boolean function by the formula

then it is easy to see that , , and for . The claim now follows from Corollary 14.

**Remark 16** The following slightly shorter variant of the argument lets one remove the factor . Let be as above. We again may normalise and . For any , let be iid Bernoulli random variable that equal with probability and with probability . The quantity

is a trigonometric polynomial of degree at most that is bounded in magnitude by , so by two applications of Bernstein’s inequality

On the other hand, for small , the random variable is equal to zero with probability and equal to each with probability , hence

and hence . Combining these estimates we obtain and hence .

**Remark 17** The sensitivity of a Boolean function can be split as , where is largest number for which there is an such that there are at least values of with . It is not difficult to use the case of Remark 3 to improve Corollary 9 slightly to . Combining this with the previous remark, we can thus improve the upper bound in Theorem 15 slightly to

### “Suite des indices de Lefschetz des itérés pour un domaine de Jordan qui est un bloc isolant”

Patrice Le Calvez and Jean-Christophe Yoccoz showed in 1997 that there are no minimal homemorphisms on the infinite annulus .

Their beautiful paper was motivated by the quest of finding minimal homeomorphisms on punctured spheres . More concretely, the non-existence of such homeomorphism was previously known when (as an easy application of the features of Lefschetz indices), (thanks to the works of Brouwer and Guillou), and (thanks to the work of Handel), so that the main result in Jean-Christophe and Patrice paper ensures the non-existence of minimal homeomorphisms in the remaining (harder) case of .

A key step in Jean-Christophe and Patrice proof of their theorem above is to establish the following result about the sequence of Lefschetz indices of iterates of a local homeomorphism of the plane at a fixed point of : if is not a sink nor a source, then there are integers such that

As it turns out, Jean-Christophe and Patrice planned a sequel to this paper with the idea of extending their techniques to compute the sequences of Lefschetz indices of periodic points of belonging to any given Jordan domain with is compact.

In fact, this plan was already known when the review of Jean-Christophe and Patrice paper came out (see here), and, as Patrice told me, some arguments from this promised subsequent work were used in the literature as a sort of folklore.

Nevertheless, a final version of this preprint was never released, and, even worse, some portions of the literature were invoking some arguments from a version of the preprint which was available only to Jean-Christophe (but not to Patrice).

Of course, this situation became slightly problematic when Jean-Christophe passed away, but fortunately Patrice and I were able to locate the final version of the preprint in Jean-Christophe’s mathematical archives. (Here, the word “final” means that all mathematical arguments are present, but the preprint has no abstract, introduction, or other “cosmetic” details.)

After doing some editing (to correct minor typos, add better figures [with the aid of Aline Cerqueira], etc.), Patrice and I are happy to announce that the folklore preprint by Jean-Christophe and Patrice (entitled “*Suite des indices de Lefschetz des itérés pour un domaine de Jordan qui est un bloc isolant*“) is finally publicly available here. We hope that you will enjoy reading this text (written in French)!

### Symmetric functions in a fractional number of variables, and the multilinear Kakeya conjecture

Let be some domain (such as the real numbers). For any natural number , let denote the space of symmetric real-valued functions on variables , thus

for any permutation . For instance, for any natural numbers , the elementary symmetric polynomials

will be an element of . With the pointwise product operation, becomes a commutative real algebra. We include the case , in which case consists solely of the real constants.

Given two natural numbers , one can “lift” a symmetric function of variables to a symmetric function of variables by the formula

where ranges over all injections from to (the latter formula making it clearer that is symmetric). Thus for instance

and

Also we have

With these conventions, we see that vanishes for , and is equal to if . We also have the transitivity

if .

The lifting map is a linear map from to , but it is not a ring homomorphism. For instance, when , one has

In general, one has the identity

for all natural numbers and , , where range over all injections , with . Combinatorially, the identity (2) follows from the fact that given any injections and with total image of cardinality , one has , and furthermore there exist precisely triples of injections , , such that and .

**Example 1** When , one has

which is just a restatement of the identity

Note that the coefficients appearing in (2) do not depend on the final number of variables . We may therefore abstract the role of from the law (2) by introducing the real algebra of formal sums

where for each , is an element of (with only finitely many of the being non-zero), and with the formal symbol being formally linear, thus

and

for and scalars , and with multiplication given by the analogue

of (2). Thus for instance, in this algebra we have

and

Informally, is an abstraction (or “inverse limit”) of the concept of a symmetric function of an unspecified number of variables, which are formed by summing terms that each involve only a bounded number of these variables at a time. One can check (somewhat tediously) that is indeed a commutative real algebra, with a unit . (I do not know if this algebra has previously been studied in the literature; it is somewhat analogous to the abstract algebra of finite linear combinations of Schur polynomials, with multiplication given by a Littlewood-Richardson rule. )

For natural numbers , there is an obvious specialisation map from to , defined by the formula

Thus, for instance, maps to and to . From (2) and (3) we see that this map is an algebra homomorphism, even though the maps and are not homomorphisms. By inspecting the component of we see that the homomorphism is in fact surjective.

Now suppose that we have a measure on the space , which then induces a product measure on every product space . To avoid degeneracies we will assume that the integral is strictly positive. Assuming suitable measurability and integrability hypotheses, a function can then be integrated against this product measure to produce a number

In the event that arises as a lift of another function , then from Fubini’s theorem we obtain the formula

is an element of the formal algebra , then

Note that by hypothesis, only finitely many terms on the right-hand side are non-zero.

Now for a key observation: whereas the left-hand side of (6) only makes sense when is a natural number, the right-hand side is meaningful when takes a fractional value (or even when it takes negative or complex values!), interpreting the binomial coefficient as a polynomial in . As such, this suggests a way to introduce a “virtual” concept of a symmetric function on a fractional power space for such values of , and even to integrate such functions against product measures , even if the fractional power does not exist in the usual set-theoretic sense (and similarly does not exist in the usual measure-theoretic sense). More precisely, for arbitrary real or complex , we now *define* to be the space of abstract objects

with and (and now interpreted as formal symbols, with the structure of a commutative real algebra inherited from , thus

In particular, the multiplication law (2) continues to hold for such values of , thanks to (3). Given any measure on , we formally define a measure on with regards to which we can integrate elements of by the formula (6) (providing one has sufficient measurability and integrability to make sense of this formula), thus providing a sort of “fractional dimensional integral” for symmetric functions. Thus, for instance, with this formalism the identities (4), (5) now hold for fractional values of , even though the formal space no longer makes sense as a set, and the formal measure no longer makes sense as a measure. (The formalism here is somewhat reminiscent of the technique of dimensional regularisation employed in the physical literature in order to assign values to otherwise divergent integrals. See also this post for an unrelated abstraction of the integration concept involving integration over supercommutative variables (and in particular over fermionic variables).)

**Example 2** Suppose is a probability measure on , and is a random variable; on any power , we let be the usual independent copies of on , thus for . Then for any real or complex , the formal integral

can be evaluated by first using the identity

(cf. (1)) and then using (6) and the probability measure hypothesis to conclude that

For a natural number, this identity has the probabilistic interpretation

whenever are jointly independent copies of , which reflects the well known fact that the sum has expectation and variance . One can thus view (7) as an abstract generalisation of (8) to the case when is fractional, negative, or even complex, despite the fact that there is no sensible way in this case to talk about independent copies of in the standard framework of probability theory.

In this particular case, the quantity (7) is non-negative for every nonnegative , which looks plausible given the form of the left-hand side. Unfortunately, this sort of non-negativity does not always hold; for instance, if has mean zero, one can check that

and the right-hand side can become negative for . This is a shame, because otherwise one could hope to start endowing with some sort of commutative von Neumann algebra type structure (or the abstract probability structure discussed in this previous post) and then interpret it as a genuine measure space rather than as a virtual one. (This failure of positivity is related to the fact that the characteristic function of a random variable, when raised to the power, need not be a characteristic function of any random variable once is no longer a natural number: “fractional convolution” does not preserve positivity!) However, one vestige of positivity remains: if is non-negative, then so is

One can wonder what the point is to all of this abstract formalism and how it relates to the rest of mathematics. For me, this formalism originated implicitly in an old paper I wrote with Jon Bennett and Tony Carbery on the multilinear restriction and Kakeya conjectures, though we did not have a good language for working with it at the time, instead working first with the case of natural number exponents and appealing to a general extrapolation theorem to then obtain various identities in the fractional case. The connection between these fractional dimensional integrals and more traditional integrals ultimately arises from the simple identity

(where the right-hand side should be viewed as the fractional dimensional integral of the unit against ). As such, one can manipulate powers of ordinary integrals using the machinery of fractional dimensional integrals. A key lemma in this regard is

**Lemma 3 (Differentiation formula)** Suppose that a positive measure on depends on some parameter and varies by the formula

for some function . Let be any real or complex number. Then, assuming sufficient smoothness and integrability of all quantities involved, we have

for all that are independent of . If we allow to now depend on also, then we have the more general total derivative formula

again assuming sufficient amounts of smoothness and regularity.

*Proof:* We just prove (10), as (11) then follows by same argument used to prove the usual product rule. By linearity it suffices to verify this identity in the case for some symmetric function for a natural number . By (6), the left-hand side of (10) is then

Differentiating under the integral sign using (9) we have

and similarly

where are the standard copies of on :

By the product rule, we can thus expand (12) as

where we have suppressed the dependence on for brevity. Since , we can write this expression using (6) as

where is the symmetric function

But from (2) one has

and the claim follows.

**Remark 4** It is also instructive to prove this lemma in the special case when is a natural number, in which case the fractional dimensional integral can be interpreted as a classical integral. In this case, the identity (10) is immediate from applying the product rule to (9) to conclude that

One could in fact derive (10) for arbitrary real or complex from the case when is a natural number by an extrapolation argument; see the appendix of my paper with Bennett and Carbery for details.

Let us give a simple PDE application of this lemma as illustration:

**Proposition 5 (Heat flow monotonicity)** Let be a solution to the heat equation with initial data a rapidly decreasing finite non-negative Radon measure, or more explicitly

for al . Then for any , the quantity

is monotone non-decreasing in for , constant for , and monotone non-increasing for .

*Proof:* By a limiting argument we may assume that is absolutely continuous, with Radon-Nikodym derivative a test function; this is more than enough regularity to justify the arguments below.

For any , let denote the Radon measure

Then the quantity can be written as a fractional dimensional integral

Observe that

and thus by Lemma 3 and the product rule

where we use for the variable of integration in the factor space of .

To simplify this expression we will take advantage of integration by parts in the variable. Specifically, in any direction , we have

and hence by Lemma 3

Multiplying by and integrating by parts, we see that

where we use the Einstein summation convention in . Similarly, if is any reasonable function depending only on , we have

and hence on integration by parts

We conclude that

and thus by (13)

The choice of that then achieves the most cancellation turns out to be (this cancels the terms that are linear or quadratic in the ), so that . Repeating the calculations establishing (7), one has

and

where is the random variable drawn from with the normalised probability measure . Since , one thus has

This expression is clearly non-negative for , equal to zero for , and positive for , giving the claim. (One could simplify here as if desired, though it is not strictly necessary to do so for the proof.)

**Remark 6** As with Remark 4, one can also establish the identity (14) first for natural numbers by direct computation avoiding the theory of fractional dimensional integrals, and then extrapolate to the case of more general values of . This particular identity is also simple enough that it can be directly established by integration by parts without much difficulty, even for fractional values of .

A more complicated version of this argument establishes the non-endpoint multilinear Kakeya inequality (without any logarithmic loss in a scale parameter ); this was established in my previous paper with Jon Bennett and Tony Carbery, but using the “natural number first” approach rather than using the current formalism of fractional dimensional integration. However, the arguments can be translated into this formalism without much difficulty; we do so below the fold. (To simplify the exposition slightly we will not address issues of establishing enough regularity and integrability to justify all the manipulations, though in practice this can be done by standard limiting arguments.)

** — 1. Multilinear heat flow monotonicity — **

Before we give a multilinear variant of Proposition 5 of relevance to the multilinear Kakeya inequality, we first need to briefly set up the theory of finite products

of fractional powers of spaces , where are real or complex numbers. The functions to integrate here lie in the tensor product space

which is generated by tensor powers

with , with the usual tensor product identifications and algebra operations. One can evaluate fractional dimensional integrals of such functions against “virtual product measures” , with a measure on , by the natural formula

assuming sufficient measurability and integrability hypotheses. We can lift functions to an element of the space (15) by the formula

This is easily seen to be an algebra homomorphism.

**Example 7** If and are functions and are measures on respectively, then (assuming sufficient measurability and integrability) then the multiple fractional dimensional integral

is equal to

In the case that are natural numbers, one can view the “virtual” integrand here as an actual function on , namely

in which case the above evaluation of the integral can be achieved classically.

From a routine application of Lemma 3 and various forms of the product rule, we see that if each varies with respect to a time parameter by the formula

and is a time-varying function in (15), then (assuming sufficient regularity and integrability), the time derivative

Now suppose that for each space one has a non-negative measure , a vector-valued function , and a matrix-valued function taking values in real symmetric positive semi-definite matrices. Let be positive real numbers; we make the abbreviations

For any and , we define the modified measures

and then the product fractional power measure

If we then define the heat-type functions

(where we drop the normalising power of for simplicity) we see in particular that

hence we can interpret the multilinear integral in the left-hand side of (17) as a product fractional dimensional integral. (We remark that in my paper with Bennett and Carbery, a slightly different parameterisation is used, replacing with , and also replacing with .)

If the functions were constant in , then the functions would obey some heat-type partial differential equation, and the situation is now very analogous to Proposition 5 (and is also closely related to Brascamp-Lieb inequalities, as discussed for instance in this paper of Carlen, Lieb, and Loss, or this paper of mine with Bennett, Carbery, and Christ). However, for applications to the multilinear Kakeya inequality, we permit to vary slightly in the variable, and now the do not directly obey any PDE.

A naive extension of Proposition 5 would then seek to establish monotonicity of the quantity (17). While such monotonicity is available in the “Brascamp-Lieb case” of constant , as discussed in the above papers, this does not quite seem to be to be true for variable . To fix this problem, a weight is introduced in order to avoid having to take matrix inverses (which are not always available in this algebra). On the product fractional dimensional space , we have a matrix-valued function defined by

The determinant is then a scalar element of the algebra (15). We then define the quantity

**Example 8** Suppose we take and let be natural numbers. Then can be viewed as the -matrix valued function

By slight abuse of notation, we write the determinant of a matrix as , where and are the first and second rows of . Then

and after some calculation, one can then write as

By a polynomial extrapolation argument, this formula is then also valid for fractional values of ; this can also be checked directly from the definitions after some tedious computation. Thus we see that while the compact-looking fractional dimensional integral (18) can be expressed in terms of more traditional integrals, the formulae get rather messy, even in the case. As such, the fractional dimensional calculus (based heavily on derivative identities such as (16)) gives a more convenient framework to manipulate these otherwise quite complicated expressions.

Suppose the functions are close to constant matrices , in the sense that

uniformly on for some small (where we use for instance the operator norm to measure the size of matrices, and we allow implied constants in the notation to depend on , and the ). Then we can write for some bounded matrix , and then we can write

We can therefore write

where and the coefficients of the matrix are some polynomial combination of the coefficients of , with all coefficients in this polynomial of bounded size. As a consequence, and on expanding out all the fractional dimensional integrals, one obtains a formula of the form

Thus, as long as is strictly positive definite and is small enough, this quantity is comparable to the classical integral

Now we compute the time derivative of . We have

so by (16), one can write as

where we use as the coordinate for the copy of that is being lifted to .

As before, we can take advantage of some cancellation in this expression using integration by parts. Since

where are the standard basis for , we see from (16) and integration by parts that

with the usual summation conventions on the index . Also, similarly to before, we suppose we have an element of (15) for each that does not depend on , then by (16) and integration by parts

or, writing ,

We can thus write (20) as

where is the element of (15) given by

The terms in that are quadratic in cancel. The linear term can be rearranged as

To cancel this, one would like to set equal to

Now in the commutative algebra (15), the inverse does not necessarily exist. However, because of the weight factor , one can work instead with the adjugate matrix , which is such that where is the identity matrix. We therefore set equal to the expression

and now the expression in (22) does not contain any linear or quadratic terms in . In particular it is completely independent of , and thus we can write

where is an arbitrary element of that we will select later to obtain a useful cancellation. We can rewrite this a little as

If we now introduce the matrix functions

and the vector functions

then this can be rewritten as

Similarly to (19), suppose that we have

uniformly on , where , thus we can write

for some bounded matrix-valued functions . Inserting this into the previous expression (and expanding out appropriately) one can eventually write

where

and is some polynomial combination of the and (or more precisely, of the quantities , , , ) that is quadratic in the variables, with bounded coefficients. As a consequence, after expanding out the product fractional dimensional integrals and applying some Cauchy-Schwarz to control cross-terms, we have

Now we simplify . We let

be the average value of ; for each this is just a vector in . We then split , leading to the identities

and

The term is problematic, but we can eliminate it as follows. By construction one has (supressing the dependence on )

By construction, one has

Thus if is positive definite and is small enough, this matrix is invertible, and we can choose so that the expression vanishes. Making this choice, we then have

Observe that the fractional dimensional integral of

or

for and arbitrary constant matrices against vanishes. As a consequence, we can now simplify the integral

Using (2), we can split

as the sum of

and

The latter also integrates to zero by the mean zero nature of . Thus we have simplified (24) to

Now let us make the key hypothesis that the matrix

is strictly positive definite, or equivalently that

for all , where the ordering is in the sense of positive definite matrices. Then we have the pointwise bound

and thus

For small enough, the expression inside the is non-negative, and we conclude the monotonicity

We have thus proven the following statement, which is essentially Proposition 4.1 of my paper with Bennett and Carbery:

**Proposition 9** Let , let be positive semi-definite real symmetric matrices, and let be such that

for . Then for any positive measure spaces with measures and any functions on with for a sufficiently small , the quantity is non-decreasing in , and is also equal to

In particular, we have

for any .

A routine calculation shows that for reasonable choices of (e.g. discrete measures of finite support), one has

and hence (setting ) we have

If we choose the to be the sum of Dirac masses, and each to be the diagonal matrix , then the key condition (25) is obeyed for , and one arrives at the multilinear Kakeya inequality

whenever are infinite tubes in of width and oriented within of the basis vector , for a sufficiently small absolute constant . (The hypothesis on the directions can then be relaxed to a transversality hypothesis by applying some linear transformations and the triangle inequality.)

### Living Proof: Stories of Resilience Along the Mathematical Journey

The AMS and MAA have recently published (and made available online) a collection of essays entitled “Living Proof: Stories of Resilience Along the Mathematical Journey”. Each author contributes a story of how they encountered some internal or external difficulty in advancing their mathematical career, and how they were able to deal with such difficulties. I myself have contributed one of these essays; I was initially somewhat surprised when I was approached for a contribution, as my career trajectory has been somewhat of an outlier, and I have been very fortunate to not experience to the same extent many of the obstacles that other contributors write about in this text. Nevertheless there was a turning point in my career that I write about here during my graduate years, when I found that the improvised and poorly disciplined study habits that were able to get me into graduate school due to an over-reliance on raw mathematical ability were completely inadequate to handle the graduate qualifying exam. With a combination of an astute advisor and some sheer luck, I was able to pass the exam and finally develop a more sustainable approach to learning and doing mathematics, but it could easily have gone quite differently. (My 20 25-year old writeup of this examination, complete with spelling errors, may be found here.)

### Yoccoz book collection at ICTP

The mathematical books of Michel Herman were donated to IMPA’s library by Jean-Christophe Yoccoz in the early 2000s: it amounts to more than 700 books and the complete list of titles can be found here.

This beautiful gesture of donating the books of a great mathematician to a developing country helped in the training of several mathematicians. In particular, I remember that reading Herman’s books during my PhD at IMPA was a singular experience in two aspects: intellectually, it gave me access to many high level mathematical topics, and olfactively, it was curious to get a smell of cigarette smoke out of old books (rather than the “usual” smell). (As I learned later, this experience was fully justified by the facts that Herman was an avid reader and a heavy smoker.)

Of course, this attitude of Jean-Christophe prompted me to discuss with Stefano Marmi about an appropriate destination in Africa to send Yoccoz’s mathematical books. After some conversations, we contacted ICTP (and, in particular, Stefano Luzzatto) to inquire about the possibility of sending Yoccoz’s books to Senegal (as a sort of “retribution” for the good memories that Jean-Christophe had during his visit to AIMS-Senegal and University of Dakar in December 2011) or Rwanda.

Unfortunately, some organisational difficulties made that we were obliged to split this plan into two parts. More concretely, rather than taking unnecessary risks by rushing to send Yoccoz’s books directly to Africa, last Thursday I sent all of them (a total of 13 boxes weighting approximately 35kg each) to ICTP library, so that they can already be useful to all ICTP visitors — in particular those coming from developing countries — instead of staying locked up in my office (where they were only sporadically read by me). In this way, we get some extra time to carefully think the definitive transfer of Yoccoz’s books to Africa while making them already publicly available.

Anyhow, the next time you visit ICTP, I hope that Yoccoz’s books will help you in some way!

### The fate of combinatorics at Strathclyde

I have just received an email from Sergey Kitaev, one of the three combinatorialists at Strathclyde. As in many universities, they belong not to the mathematics department but to the computer science department. Kitaev informs me that the administrators of that department, in their infinite wisdom, have decided that the future of the department is best served by axing discrete mathematics. I won’t write a long post about this, but instead refer you to a post by Peter Cameron that says everything I would want to say about the decision, and does so extremely cogently. I recommend that you read it if this kind of decision worries you.

### Abstracting induction on scales arguments

The following situation is very common in modern harmonic analysis: one has a large scale parameter (sometimes written as in the literature for some small scale parameter , or as for some large radius ), which ranges over some unbounded subset of (e.g. all sufficiently large real numbers , or all powers of two), and one has some positive quantity depending on that is known to be of *polynomial size* in the sense that

for all in the range and some constant , and one wishes to obtain a *subpolynomial upper bound* for , by which we mean an upper bound of the form

for all and all in the range, where can depend on but is independent of . In many applications, this bound is nearly tight in the sense that one can easily establish a matching lower bound

in which case the property of having a subpolynomial upper bound is equivalent to that of being *subpolynomial size* in the sense that

for all and all in the range. It would naturally be of interest to tighten these bounds further, for instance to show that is polylogarithmic or even bounded in size, but a subpolynomial bound is already sufficient for many applications.

Let us give some illustrative examples of this type of problem:

**Example 1 (Kakeya conjecture)** Here ranges over all of . Let be a fixed dimension. For each , we pick a maximal -separated set of directions . We let be the smallest constant for which one has the Kakeya inequality

where is a -tube oriented in the direction . The Kakeya maximal function conjecture is then equivalent to the assertion that has a subpolynomial upper bound (or equivalently, is of subpolynomial size). Currently this is only known in dimension .

**Example 2 (Restriction conjecture for the sphere)** Here ranges over all of . Let be a fixed dimension. We let be the smallest constant for which one has the restriction inequality

for all bounded measurable functions on the unit sphere equipped with surface measure , where is the ball of radius centred at the origin. The restriction conjecture of Stein for the sphere is then equivalent to the assertion that has a subpolynomial upper bound (or equivalently, is of subpolynomial size). Currently this is only known in dimension .

**Example 3 (Multilinear Kakeya inequality)** Again ranges over all of . Let be a fixed dimension, and let be compact subsets of the sphere which are *transverse* in the sense that there is a uniform lower bound for the wedge product of directions for (equivalently, there is no hyperplane through the origin that intersects all of the ). For each , we let be the smallest constant for which one has the multilinear Kakeya inequality

where for each , is a collection of infinite tubes in of radius oriented in a direction in , which are separated in the sense that for any two tubes in , either the directions of differ by an angle of at least , or are disjoint; and is our notation for the geometric mean

The multilinear Kakeya inequality of Bennett, Carbery, and myself establishes that is of subpolynomial size; a later argument of Guth improves this further by showing that is bounded (and in fact comparable to ).

**Example 4 (Multilinear restriction theorem)** Once again ranges over all of . Let be a fixed dimension, and let be compact subsets of the sphere which are *transverse* as in the previous example. For each , we let be the smallest constant for which one has the multilinear restriction inequality

for all bounded measurable functions on for . Then the multilinear restriction theorem of Bennett, Carbery, and myself establishes that is of subpolynomial size; it is known to be bounded for (as can be easily verified from Plancherel’s theorem), but it remains open whether it is bounded for any .

**Example 5 (Decoupling for the paraboloid)** now ranges over the square numbers. Let , and subdivide the unit cube into cubes of sidelength . For any , define the extension operators

and

for and . We also introduce the weight function

For any , let be the smallest constant for which one has the decoupling inequality

The decoupling theorem of Bourgain and Demeter asserts that is of subpolynomial size for all in the optimal range .

**Example 6 (Decoupling for the moment curve)** now ranges over the natural numbers. Let , and subdivide into intervals of length . For any , define the extension operators

and more generally

for . For any , let be the smallest constant for which one has the decoupling inequality

It was shown by Bourgain, Demeter, and Guth that is of subpolynomial size for all in the optimal range , which among other things implies the Vinogradov main conjecture (as discussed in this previous post).

It is convenient to use asymptotic notation to express these estimates. We write , , or to denote the inequality for some constant independent of the scale parameter , and write for . We write to denote a bound of the form where as along the given range of . We then write for , and for . Then the statement that is of polynomial size can be written as

while the statement that has a subpolynomial upper bound can be written as

and similarly the statement that is of subpolynomial size is simply

Many modern approaches to bounding quantities like in harmonic analysis rely on some sort of *induction on scales* approach in which is bounded using quantities such as for some exponents . For instance, suppose one is somehow able to establish the inequality

for all , and suppose that is also known to be of polynomial size. Then this implies that has a subpolynomial upper bound. Indeed, one can iterate this inequality to show that

for any fixed ; using the polynomial size hypothesis one thus has

for some constant independent of . As can be arbitrarily large, we conclude that for any , and hence is of subpolynomial size. (This sort of iteration is used for instance in my paper with Bennett and Carbery to derive the multilinear restriction theorem from the multilinear Kakeya theorem.)

**Exercise 7** If is of polynomial size, and obeys the inequality

for any fixed , where the implied constant in the notation is independent of , show that has a subpolynomial upper bound. This type of inequality is used to equate various linear estimates in harmonic analysis with their multilinear counterparts; see for instance this paper of myself, Vargas, and Vega for an early example of this method.

In more recent years, more sophisticated induction on scales arguments have emerged in which one or more auxiliary quantities besides also come into play. Here is one example, this time being an abstraction of a short proof of the multilinear Kakeya inequality due to Guth. Let be the quantity in Example 3. We define similarly to for any , except that we now also require that the diameter of each set is at most . One can then observe the following estimates:

- (Triangle inequality) For any , we have
- (Multiplicativity) For any , one has
- (Loomis-Whitney inequality) We have

These inequalities now imply that has a subpolynomial upper bound, as we now demonstrate. Let be a large natural number (independent of ) to be chosen later. From many iterations of (6) we have

and hence by (7) (with replaced by ) and (5)

where the implied constant in the exponent does not depend on . As can be arbitrarily large, the claim follows. We remark that a nearly identical scheme lets one deduce decoupling estimates for the three-dimensional cone from that of the two-dimensional paraboloid; see the final section of this paper of Bourgain and Demeter.

Now we give a slightly more sophisticated example, abstracted from the proof of decoupling of the paraboloid by Bourgain and Demeter, as described in this study guide after specialising the dimension to and the exponent to the endpoint (the argument is also more or less summarised in this previous post). (In the cited papers, the argument was phrased only for the non-endpoint case , but it has been observed independently by many experts that the argument extends with only minor modifications to the endpoint .) Here we have a quantity that we wish to show is of subpolynomial size. For any and , one can define an auxiliary quantity . The precise definitions of and are given in the study guide (where they are called and respectively, setting and ) but will not be of importance to us for this discussion. Suffice to say that the following estimates are known:

- (Crude upper bound for ) is of polynomial size: .
- (Bilinear reduction, using parabolic rescaling) For any , one has
- (Crude upper bound for ) For any one has
- (Application of multilinear Kakeya and decoupling) If are sufficiently small (e.g. both less than ), then

In all of these bounds the implied constant exponents such as or are independent of and , although the implied constants in the notation can depend on both and . Here we gloss over an annoying technicality in that quantities such as , , or might not be an integer (and might not divide evenly into ), which is needed for the application to decoupling theorems; this can be resolved by restricting the scales involved to powers of two and restricting the values of to certain rational values, which introduces some complications to the later arguments below which we shall simply ignore as they do not significantly affect the numerology.

It turns out that these estimates imply that is of subpolynomial size. We give the argument as follows. As is known to be of polynomial size, we have some for which we have the bound

for all . We can pick to be the minimal exponent for which this bound is attained: thus

We will call this the *upper exponent* of . We need to show that . We assume for contradiction that . Let be a sufficiently small quantity depending on to be chosen later. From (10) we then have

for any sufficiently small . A routine iteration then gives

for any that is independent of , if is sufficiently small depending on . A key point here is that the implied constant in the exponent is uniform in (the constant comes from summing a convergent geometric series). We now use the crude bound (9) followed by (11) and conclude that

Applying (8) we then have

If we choose sufficiently large depending on (which was assumed to be positive), then the negative term will dominate the term. If we then pick sufficiently small depending on , then finally sufficiently small depending on all previous quantities, we will obtain for some strictly less than , contradicting the definition of . Thus cannot be positive, and hence has a subpolynomial upper bound as required.

**Exercise 8** Show that one still obtains a subpolynomial upper bound if the estimate (10) is replaced with

for some constant , so long as we also improve (9) to

(This variant of the argument lets one handle the non-endpoint cases of the decoupling theorem for the paraboloid.)

To establish decoupling estimates for the moment curve, restricting to the endpoint case for sake of discussion, an even more sophisticated induction on scales argument was deployed by Bourgain, Demeter, and Guth. The proof is discussed in this previous blog post, but let us just describe an abstract version of the induction on scales argument. To bound the quantity , some auxiliary quantities are introduced for various exponents and and , with the following bounds:

- (Crude upper bound for ) is of polynomial size: .
- (Multilinear reduction, using non-isotropic rescaling) For any and , one has
- (Crude upper bound for ) For any and one has
- (Hölder) For and one has
- (Rescaled decoupling hypothesis) For , one has
- (Lower dimensional decoupling) If and , then
- (Multilinear Kakeya) If and , then

It is now substantially less obvious that these estimates can be combined to demonstrate that is of subpolynomial size; nevertheless this can be done. A somewhat complicated arrangement of the argument (involving some rather unmotivated choices of expressions to induct over) appears in my previous blog post; I give an alternate proof later in this post.

These examples indicate a general strategy to establish that some quantity is of subpolynomial size, by

- (i) Introducing some family of related auxiliary quantities, often parameterised by several further parameters;
- (ii) establishing as many bounds between these quantities and the original quantity as possible; and then
- (iii) appealing to some sort of “induction on scales” to conclude.

The first two steps (i), (ii) depend very much on the harmonic analysis nature of the quantities and the related auxiliary quantities, and the estimates in (ii) will typically be proven from various harmonic analysis inputs such as Hölder’s inequality, rescaling arguments, decoupling estimates, or Kakeya type estimates. The final step (iii) requires no knowledge of where these quantities come from in harmonic analysis, but the iterations involved can become extremely complicated.

In this post I would like to observe that one can clean up and made more systematic this final step (iii) by passing to upper exponents (12) to eliminate the role of the parameter (and also “tropicalising” all the estimates), and then taking similar limit superiors to eliminate some other less important parameters, until one is left with a simple linear programming problem (which, among other things, could be amenable to computer-assisted proving techniques). This method is analogous to that of passing to a simpler asymptotic limit object in many other areas of mathematics (for instance using the Furstenberg correspondence principle to pass from a combinatorial problem to an ergodic theory problem, as discussed in this previous post). We use the limit superior exclusively in this post, but many of the arguments here would also apply with one of the other generalised limit functionals discussed in this previous post, such as ultrafilter limits.

For instance, if is the upper exponent of a quantity of polynomial size obeying (4), then a comparison of the upper exponent of both sides of (4) one arrives at the scalar inequality

from which it is immediate that , giving the required subpolynomial upper bound. Notice how the passage to upper exponents converts the estimate to a simpler inequality .

**Exercise 9** Repeat Exercise 7 using this method.

Similarly, given the quantities obeying the axioms (5), (6), (7), and assuming that is of polynomial size (which is easily verified for the application at hand), we see that for any real numbers , the quantity is also of polynomial size and hence has some upper exponent ; meanwhile itself has some upper exponent . By reparameterising we have the homogeneity

for any . Also, comparing the upper exponents of both sides of the axioms (5), (6), (7) we arrive at the inequalities

For any natural number , the third inequality combined with homogeneity gives , which when combined with the second inequality gives , which on combination with the first estimate gives . Sending to infinity we obtain as required.

Now suppose that , obey the axioms (8), (9), (10). For any fixed , the quantity is of polynomial size (thanks to (9) and the polynomial size of ), and hence has some upper exponent ; similarly has some upper exponent . (Actually, strictly speaking our axioms only give an upper bound on so we have to temporarily admit the possibility that , though this will soon be eliminated anyway.) Taking upper exponents of all the axioms we then conclude that

for all and .

Assume for contradiction that , then , and so the statement (20) simplifies to

At this point we can eliminate the role of and simplify the system by taking a second limit superior. If we write

then on taking limit superiors of the previous inequalities we conclude that

for all ; in particular . We take advantage of this by taking a further limit superior (or “upper derivative”) in the limit to eliminate the role of and simplify the system further. If we define

so that is the best constant for which as , then is finite, and by inserting this “Taylor expansion” into the right-hand side of (21) and conclude that

This leads to a contradiction when , and hence as desired.

**Exercise 10** Redo Exercise 8 using this method.

The same strategy now clarifies how to proceed with the more complicated system of quantities obeying the axioms (13)–(19) with of polynomial size. Let be the exponent of . From (14) we see that for fixed , each is also of polynomial size (at least in upper bound) and so has some exponent (which for now we can permit to be ). Taking upper exponents of all the various axioms we can now eliminate and arrive at the simpler axioms

for all , , and , with the lower dimensional decoupling inequality

for and , and the multilinear Kakeya inequality

for and .

As before, if we assume for sake of contradiction that then the first inequality simplifies to

We can then again eliminate the role of by taking a second limit superior as , introducing

and thus getting the simplified axiom system

for and , and

for and .

In view of the latter two estimates it is natural to restrict attention to the quantities for . By the axioms (22), these quantities are of the form . We can then eliminate the role of by taking another limit superior

The axioms now simplify to

It turns out that the inequality (27) is strongest when , thus

From the last two inequalities (28), (29) we see that a special role is likely to be played by the exponents

for and

for . From the convexity (25) and a brief calculation we have

for , hence from (28) we have

Similarly, from (25) and a brief calculation we have

for ; the same bound holds for if we drop the term with the factor, thanks to (24). Thus from (29) we have

for , again with the understanding that we omit the first term on the right-hand side when . Finally, (26) gives

Let us write out the system of equations we have obtained in full:

We can then eliminate the variables one by one. Inserting (33) into (32) we obtain

which simplifies to

Inserting this into (34) gives

which when combined with (35) gives

which simplifies to

Iterating this we get

for all and

for all . In particular

which on insertion into (36), (37) gives

which is absurd if . Thus and so must be of subpolynomial growth.

**Remark 11** (This observation is essentially due to Heath-Brown.) If we let denote the column vector with entries (arranged in whatever order one pleases), then the above system of inequalities (32)–(36) (using (37) to handle the appearance of in (36)) reads

for some explicit square matrix with non-negative coefficients, where the inequality denotes pointwise domination, and is an explicit vector with non-positive coefficients that reflects the effect of (37). It is possible to show (using (24), (26)) that all the coefficients of are negative (assuming the counterfactual situation of course). Then we can iterate this to obtain

for any natural number . This would lead to an immediate contradiction if the Perron-Frobenius eigenvalue of exceeds because would now grow exponentially; this is typically the situation for “non-endpoint” applications such as proving decoupling inequalities away from the endpoint. In the endpoint situation discussed above, the Perron-Frobenius eigenvalue is , with having a non-trivial projection to this eigenspace, so the sum now grows at least linearly, which still gives the required contradiction for any . So it is important to gather “enough” inequalities so that the relevant matrix has a Perron-Frobenius eigenvalue greater than or equal to (and in the latter case one needs non-trivial injection of an induction hypothesis into an eigenspace corresponding to an eigenvalue ). More specifically, if is the spectral radius of and is a left Perron-Frobenius eigenvector, that is to say a non-negative vector, not identically zero, such that , then by taking inner products of (38) with we obtain

If this leads to a contradiction since is negative and is non-positive. When one still gets a contradiction as long as is strictly negative.

**Remark 12** (This calculation is essentially due to Guo and Zorin-Kranich.) Here is a concrete application of the Perron-Frobenius strategy outlined above to the system of inequalities (32)–(37). Consider the weighted sum

I had secretly calculated the weights , as coming from the left Perron-Frobenius eigenvector of the matrix described in the previous remark, but for this calculation the precise provenance of the weights is not relevant. Applying the inequalities (31), (30) we see that is bounded by

(with the convention that the term is absent); this simplifies after some calculation to the bound

and this and (37) then leads to the required contradiction.

**Exercise 13**

- (i) Extend the above analysis to also cover the non-endpoint case . (One will need to establish the claim for .)
- (ii) Modify the argument to deal with the remaining cases by dropping some of the steps.

### Ruling out polynomial bijections over the rationals via Bombieri-Lang?

I recently came across this question on MathOverflow asking if there are any polynomials of two variables with rational coefficients, such that the map is a bijection. The answer to this question is almost surely “no”, but it is remarkable how hard this problem resists any attempt at rigorous proof. (MathOverflow users with enough privileges to see deleted answers will find that there are no fewer than seventeen deleted attempts at a proof in response to this question!)

On the other hand, the one surviving response to the question does point out this paper of Poonen which shows that assuming a powerful conjecture in Diophantine geometry known as the Bombieri-Lang conjecture (discussed in this previous post), it is at least possible to exhibit polynomials which are injective.

I believe that it should be possible to also rule out the existence of bijective polynomials if one assumes the Bombieri-Lang conjecture, and have sketched out a strategy to do so, but filling in the gaps requires a fair bit more algebraic geometry than I am capable of. So as a sort of experiment, I would like to see if a rigorous implication of this form (similarly to the rigorous implication of the Erdos-Ulam conjecture from the Bombieri-Lang conjecture in my previous post) can be crowdsourced, in the spirit of the polymath projects (though I feel that this particular problem should be significantly quicker to resolve than a typical such project).

Here is how I imagine a Bombieri-Lang-powered resolution of this question should proceed (modulo a large number of unjustified and somewhat vague steps that I believe to be true but have not established rigorously). Suppose for contradiction that we have a bijective polynomial . Then for any polynomial of one variable, the surface

has infinitely many rational points; indeed, every rational lifts to exactly one rational point in . I believe that for “typical” this surface should be irreducible. One can now split into two cases:

- (a) The rational points in are Zariski dense in .
- (b) The rational points in are not Zariski dense in .

Consider case (b) first. By definition, this case asserts that the rational points in are contained in a finite number of algebraic curves. By Faltings’ theorem (a special case of the Bombieri-Lang conjecture), any curve of genus two or higher only contains a finite number of rational points. So all but finitely many of the rational points in are contained in a finite union of genus zero and genus one curves. I think all genus zero curves are birational to a line, and all the genus one curves are birational to an elliptic curve (though I don’t have an immediate reference for this). These curves all can have an infinity of rational points, but very few of them should have “enough” rational points that their projection to the third coordinate is “large”. In particular, I believe

- (i) If is birational to an elliptic curve, then the number of elements of of height at most should grow at most polylogarithmically in (i.e., be of order .
- (ii) If is birational to a line but not of the form for some rational , then then the number of elements of of height at most should grow slower than (in fact I think it can only grow like ).

I do not have proofs of these results (though I think something similar to (i) can be found in Knapp’s book, and (ii) should basically follow by using a rational parameterisation of with nonlinear). Assuming these assertions, this would mean that there is a curve of the form that captures a “positive fraction” of the rational points of , as measured by restricting the height of the third coordinate to lie below a large threshold , computing density, and sending to infinity (taking a limit superior). I believe this forces an identity of the form

for all . Such identities are certainly possible for some choices of (e.g. for arbitrary polynomials of one variable) but I believe that the only way that such identities hold for a “positive fraction” of (as measured using height as before) is if there is in fact a rational identity of the form

for some rational functions with rational coefficients (in which case we would have and ). But such an identity would contradict the hypothesis that is bijective, since one can take a rational point outside of the curve , and set , in which case we have violating the injective nature of . Thus, modulo a lot of steps that have not been fully justified, we have ruled out the scenario in which case (b) holds for a “positive fraction” of .

This leaves the scenario in which case (a) holds for a “positive fraction” of . Assuming the Bombieri-Lang conjecture, this implies that for such , any resolution of singularities of fails to be of general type. I would imagine that this places some very strong constraints on , since I would expect the equation to describe a surface of general type for “generic” choices of (after resolving singularities). However, I do not have a good set of techniques for detecting whether a given surface is of general type or not. Presumably one should proceed by viewing the surface as a fibre product of the simpler surface and the curve over the line . In any event, I believe the way to handle (a) is to show that the failure of general type of implies some strong algebraic constraint between and (something in the spirit of (1), perhaps), and then use this constraint to rule out the bijectivity of by some further *ad hoc* method.

### Dynamics on character varieties: Brown’s theorem revisited

Let be a surface of genus with punctures. Given a Lie group , the -character variety of is the space of representations modulo conjugations by elements of .

The mapping class group of isotopy classes of orientation-preserving diffeomorphisms of acts naturally on .

The dynamics of mapping class groups on character varieties was systematically studied by Goldman in 1997: in his landmark paper, he showed that the -action on is ergodic with respect to Goldman–Huebschmann measure whenever .

**Remark 1** *This nomenclature is not standard: we use it here because Goldman showed here that has a volume form coming from a natural symplectic structure and Huebschmann proved here that this volume form has finite mass.*

The ergodicity result above partly motivates the question of understanding the dynamics of individual elements of mapping class groups acting on -character varieties.

In this direction, Brown studied in 1998 the actions of elements of on the character variety . As it turns out, if is a small loop around the puncture, then the -action on preserves each level set , , of the function sending to the trace of the matrix . Here, Brown noticed that the dynamics of elements of on level sets with close to fit the setting of the celebrated KAM theory (assuring the stability of non-degenerate elliptic periodic points of smooth area-preserving maps). In particular, Brown tried to employ Moser’s twisting theorem to conclude that no element of can act ergodically on all level sets , .

Strictly speaking, Brown’s original argument is not complete because Moser’s theorem is used without checking the twist condition.

In the sequel, we revisit Brown’s work in order to show that his conclusions can be derived once one replaces Moser’s twisting theorem by a KAM stability theorem from 2002 due to Rüssmann.

**1. Statement of Brown’s theorem**

**1.1. -character variety of a punctured torus**

Recall that the fundamental group of an once-punctured torus is naturally isomorphic to a free group on two generators and such that the commutator corresponds to a loop around the puncture of .

Therefore, a representation is determined by a pair of matrices , and an element of the -character variety of is determined by the simultaneous conjugacy class , , of a pair of matrices .

The traces , and of the matrices , and provide an useful system of coordinates on : algebraically, this is an incarnation of the fact that the ring of invariants of is freely generated by the traces of , and .

In particular, the following proposition expresses the trace of in terms of , and .

*Proof:* By Cayley–Hamilton theorem (or a direct calculation), any satisfies , i.e., .

Hence, for any , one has

so that

It follows that, for any , one has

and

Since and , the proof of the proposition is complete.

**1.2. Basic dynamics of on character varieties**

Recall that the mapping class group is generated by Dehn twists and about the generators and of . In appropriate coordinates on the once-punctured torus , the isotopy classes of these Dehn twists are represented by the actions of the matrices

on the flat torus . In particular, at the homotopy level, the actions of and on are given by the Nielsen transformations

Since the elements of fix the puncture of , they preserve the homotopy class of a small loop around the puncture. Therefore, the -action on the character variety respects the level sets , , of the function given by

Furthermore, each level set , , carries a finite (*Goldman*—*Huebschmann*) measure coming from a natural -invariant symplectic structure.

In this context, the level set corresponds to impose the restriction , so that is naturally identified with the character variety .

In terms of the coordinates , and on , we can use Proposition 1 (and its proof) and (1) to check that

Hence, we see from (2) that:

- the level set consists of a single point ;
- the level sets , , are diffeomorphic to -spheres;
- the character variety is a -dimensional orbifold whose boundary is a topological sphere with 4 singular points (of coordinates with ) corresponding to the character variety .

After this brief discussion of some geometrical aspects of , we are ready to begin the study of the dynamics of . For this sake, recall that the elements of are classified into three types:

- is called elliptic whenever ;
- is called parabolic whenever ;
- is hyperbolic whenever .

The elliptic elements have finite order (because and ) and the parabolic elements are conjugated to for some .

In particular, if is elliptic, then leaves invariant non-trivial open subsets of each level set , . Moreover, if is parabolic, then preserves a non-trivial and non-peripheral element and, *a fortiori*, preserves the level sets of the function , . Since any such function has a non-constant restriction to any level set , , Brown concluded that:

**Proposition 2 (Proposition 4.3 of Brown’s paper)** *If is not hyperbolic, then its action on is not ergodic whenever .*

On the other hand, Brown observed that the action of any hyperbolic element of on can be understood via a result of Katok.

**Proposition 3 (Theorem 4.1 of Brown’s paper)** *Any hyperbolic element of acts ergodically on .*

*Proof:* The level set is the character variety . In other words, a point in represents the simultaneous conjugacy class of a pair of *commuting* matrices in .

Since a maximal torus of is a conjugate of the subgroup

we have that is the set of simultaneous conjugacy classes of elements of . In view of the action by conjugation

of the element of the Weyl subgroup of , we have

In terms of the coordinates given by the phases of the elements

the element acts by , so that is the topological sphere obtained from the quotient of by its hyperelliptic involution (and has only four singular points located at the subset of fixed points of the hyperelliptic involution). Moreover, an element acts on by mapping to .

In summary, the action of on is given by the usual -action on the topological sphere induced from the standard on the torus .

By a result of Katok, it follows that the action of any hyperbolic element of on is ergodic (and actually Bernoulli).

**1.3. Brown’s theorem**

The previous two propositions raise the question of the ergodicity of the action of hyperbolic elements of on the level sets , . The following theorem of Brown provides an answer to this question:

**Theorem 4** *Let be an hyperbolic element of . Then, there exists such that does not act ergodically on .*

Very roughly speaking, Brown establishes Theorem 4 along the following lines. One starts by performing a blowup at the origin in order to think of the action of on as a one-parameter family , , of area-preserving maps of the -sphere such that is a finite order element of . In this way, we have that is a non-trivial one-parameter family going from a completely elliptic behaviour at to a non-uniformly hyperbolic behaviour at . This scenario suggests that the conclusion of Theorem 4 can be derived via KAM theory in the elliptic regime.

In the next (and last) section of this post, we revisit Brown’s ideas leading to Theorem 4 (with an special emphasis on its KAM theoretical aspects).

**2. Revisited proof of Brown’s theorem**

**2.1. Blowup of the origin**

The origin of the character variety can be blown up into a sphere of directions . The action of on factors through an octahedral subgroup of : this follows from the fact that (3) implies that the generators and of act on as

In this way, each element is related to a root of unity

of order coming from the eigenvalues of the derivative of at any of its fixed points.

**Example 1** *The hyperbolic element acts on via the element of of order .*

**2.2. Bifurcations of fixed points**

An hyperbolic element induces a non-trivial polynomial automorphism of whose restriction to describe the action of on . In particular, the set of fixed points of this polynomial automorphism in is a semi-algebraic set of dimension .

Actually, it is not hard to exploit the fact that acts on the level sets , , through area-preserving maps to compute the Zariski tangent space to in order to verify that is one-dimensional (cf. Proposition 5.1 in Brown’s work).

Moreover, this calculation of Zariski tangent space can be combined with the fact that any hyperbolic element has a discrete set of fixed points in and, a fortiori, in to get that is transverse to except at its discrete subset of singular points and, hence, is discrete for all (cf. Proposition 5.2 in Brown’s work).

**Example 2** *The hyperbolic element acts on via the polynomial automorphism (cf. (3)). Thus, the corresponding set of fixed points is given by the equations*

*describing an embedded curve in .*

In general, the eigenvalues of the derivative at of the action of an hyperbolic element on can be continuously followed along any irreducible component of .

Furthermore, it is not hard to check that is not constant on (cf. Lemma 5.3 in Brown’s work). Indeed, this happens because there are only two cases: the first possibility is that connects and so that varies from to the unstable eigenvalue of acting on ; the second possibility is that becomes tangent to for some so that the Zariski tangent space computation mentioned above reveals that varies from (at ) to some value (at any point of transverse intersection between and a level set of ).

**2.3. Detecting Brjuno elliptic periodic points**

The discussion of the previous two subsections allows to show that the some portions of the action of an hyperbolic element fit the assumptions of KAM theory.

Before entering into this matter, recall that is Brjuno whenever is an irrational number whose continued fraction has partial convergents satisfying

For our purposes, it is important to note that the Brjuno condition has full Lebesgue measure on .

Let be an hyperbolic element. We have three possibilities for the limiting eigenvalue : it is not real, it equals or it equals .

If the limiting eigenvalue is not real, then we take an irreducible component intersecting the origin . Since is not constant on implies that contains an open subset of . Thus, we can find some such that has a Brjuno eigenvalue , i.e., the action of on has a Brjuno fixed point.

If the limiting eigenvalue is , we use Lefschetz fixed point theorem on the sphere with close to to locate an irreducible component of such that is a fixed point of positive index of for close to . On the other hand, it is known that an isolated fixed point of an orientation-preserving surface homeomorphism which preserves area has index . Therefore, is a fixed point of of index with multipliers close to whenever is close to . Since a hyperbolic fixed point with positive multipliers has index , it follows that is a fixed point with when is close to . In particular, contains an open subset of and, hence, we can find some such that has a Brjuno multiplier .

If the limiting eigenvalue is , then is an hyperbolic element with limiting eigenvalue . From the previous paragraph, it follows that we can find some such that contains a Brjuno elliptic fixed point of .

In any event, the arguments above give the following result (cf. Theorem 4.4 in Brown’s work):

**Theorem 5** *Let be an hyperbolic element. Then, there exists such that has a periodic point of period one or two with a Brjuno multiplier.*

**2.4. Moser’s twisting theorem and Rüssmann’s stability theorem**

At this point, the idea to derive Theorem 4 is to combine Theorem 5 with KAM theory ensuring the stability of certain types of elliptic periodic points.

Recall that a periodic point is called stable whenever there are arbitrarily small neighborhoods of its orbit which are invariant. In particular, the presence of a stable periodic point implies the non-ergodicity of an area-preserving map.

A famous stability criterion for fixed points of area-preserving maps is Moser’s twisting theorem. This result can be stated as follows. Suppose that is an area-preserving , , map having an elliptic fixed point at origin with multipliers , such that for . After performing an appropriate area-preserving change of variables (tangent to the identity at the origin), one can bring into its Birkhoff normal form, i.e., has the form

where , , are uniquely determined Birkhoff constants and denotes higher order terms.

**Theorem 6 (Moser twisting theorem)** *Let be an area-preserving map as in the previous paragraph. If for some , then the origin is a stable fixed point.*

The nomenclature “twisting” comes from the fact when is a twist map, i.e., has the form in polar coordinates where is a smooth function with . In the literature, the condition “ for some ” is called twist condition.

**Example 3** *The Dehn twist induces the polynomial automorphism on . Each level set , , is a smooth -sphere which is swept out by the -invariant ellipses obtained from the intersections between and the planes of the form .**Goldman observed that, after an appropriate change of coordinates, each becomes a circle where acts as a rotation by angle . In particular, the restriction of to each level set is a twist map near its fixed points .*

In his original argument, Brown deduced Theorem 4 from (a weaker version of) Theorem 5 and Moser’s twisting theorem. However, Brown employed Moser’s theorem with while checking only the conditions on the multipliers of the elliptic fixed point but not the twist condition .

As it turns out, it is not obvious to check the twist condition in Brown’s setting (especially because it is not satisfied at the sphere of directions ).

Fortunately, Rüssmann discovered that a Brjuno elliptic fixed point of a real-analytic area-preserving map is always stable (independently of twisting conditions):

**Theorem 7 (Rüssmann)** *Any Brjuno elliptic periodic point of a real-analytic area-preserving map is stable.*

**Remark 2** *Actually, Rüssmann obtained the previous result by showing that a real-analytic area-preserving map with a Brjuno elliptic fixed point and vanishing Birkhoff constants (i.e., for all ) is analytically linearisable. Note that the analogue of this statement in the category is false (as a counterexample is given by ).*

In any case, at this stage, the proof of Theorem 4 is complete: it suffices to put together Theorems 5 and 7.

### Searching for singularities in the Navier–Stokes equations

I was recently asked to contribute a short comment to Nature Reviews Physics, as part of a series of articles on fluid dynamics on the occasion of the 200th anniversary (this August) of the birthday of George Stokes. My contribution is now online as “Searching for singularities in the Navier–Stokes equations“, where I discuss the global regularity problem for Navier-Stokes and my thoughts on how one could try to construct a solution that blows up in finite time via an approximately discretely self-similar “fluid computer”. (The rest of the series does not currently seem to be available online, but I expect they will become so shortly.)

### The spherical Cayley-Menger determinant and the radius of the Earth

Given three points in the plane, the distances between them have to be non-negative and obey the triangle inequalities

but are otherwise unconstrained. But if one has *four* points in the plane, then there is an additional constraint connecting the six distances between them, coming from the Cayley-Menger determinant:

**Proposition 1 (Cayley-Menger determinant)** If are four points in the plane, then the Cayley-Menger determinant

*Proof:* If we view as vectors in , then we have the usual cosine rule , and similarly for all the other distances. The matrix appearing in (1) can then be written as , where is the matrix

and is the (augmented) Gram matrix

The matrix is a rank one matrix, and so is also. The Gram matrix factorises as , where is the matrix with rows , and thus has rank at most . Therefore the matrix in (1) has rank at most , and hence has determinant zero as claimed.

For instance, if we know that and , then in order for to be coplanar, the remaining distance has to obey the equation

After some calculation the left-hand side simplifies to , so the non-negative quantity is constrained to equal either or . The former happens when form a unit right-angled triangle with right angle at and ; the latter happens when form the vertices of a unit square traversed in that order. Any other value for is not compatible with the hypothesis for lying on a plane; hence the Cayley-Menger determinant can be used as a test for planarity.

Now suppose that we have four points on a sphere of radius , with six distances now measured as lengths of arcs on the sphere. There is a spherical analogue of the Cayley-Menger determinant:

**Proposition 2 (Spherical Cayley-Menger determinant)** If are four points on a sphere of radius in , then the spherical Cayley-Menger determinant

*Proof:* We can assume that the sphere is centred at the origin of , and view as vectors in of magnitude . The angle subtended by from the origin is , so by the cosine rule we have

Similarly for all the other inner products. Thus the matrix in (2) can be written as , where is the Gram matrix

We can factor where is the matrix with rows . Thus has rank at most and thus the determinant vanishes as required.

Just as the Cayley-Menger determinant can be used to test for coplanarity, the spherical Cayley-Menger determinant can be used to test for lying on a sphere of radius . For instance, if we know that lie on and are all equal to , then the above proposition gives

The left-hand side evaluates to ; as lies between and , the only choices for this distance are then and . The former happens for instance when lies on the north pole , are points on the equator with longitudes differing by 90 degrees, and is also equal to the north pole; the latter occurs when is instead placed on the south pole.

The Cayley-Menger and spherical Cayley-Menger determinants look slightly different from each other, but one can transform the latter into something resembling the former by row and column operations. Indeed, the determinant (2) can be rewritten as

and by further row and column operations, this determinant vanishes if and only if the determinant

vanishes, where . In the limit (so that the curvature of the sphere tends to zero), tends to , and by Taylor expansion tends to ; similarly for the other distances. Now we see that the planar Cayley-Menger determinant emerges as the limit of (3) as , as would be expected from the intuition that a plane is essentially a sphere of infinite radius.

In principle, one can now estimate the radius of the Earth (assuming that it is either a sphere or a flat plane ) if one is given the six distances between four points on the Earth. Of course, if one wishes to do so, one should have rather far apart from each other, since otherwise it would be difficult to for instance distinguish the round Earth from a flat one. As an experiment, and just for fun, I wanted to see how accurate this would be with some real world data. I decided to take , , , be the cities of London, Los Angeles, Tokyo, and Dubai respectively. As an initial test, I used distances from this online flight calculator, measured in kilometers:

Given that the true radius of the earth was about kilometers, I chose the change of variables (so that corresponds to the round Earth model with the commonly accepted value for the Earth’s radius, and corresponds to the flat Earth), and obtained the following plot for (3):

In particular, the determinant does indeed come very close to vanishing when , which is unsurprising since, as explained on the web site, the online flight calculator uses a model in which the Earth is an ellipsoid of radii close to km. There is another radius that would also be compatible with this data at (corresponding to an Earth of radius about km), but presumably one could rule out this as a spurious coincidence by experimenting with other quadruples of cities than the ones I selected. On the other hand, these distances are highly incompatible with the flat Earth model ; one could also see this with a piece of paper and a ruler by trying to lay down four points on the paper with (an appropriately rescaled) version of the above distances (e.g., with , , etc.).

If instead one goes to the flight time calculator and uses flight travel times instead of distances, one now gets the following data (measured in hours):

Assuming that planes travel at about kilometers per hour, the true radius of the Earth should be about of flight time. If one then uses the normalisation , one obtains the following plot:

Not too surprisingly, this is basically a rescaled version of the previous plot, with vanishing near and at . (The website for the flight calculator does say it calculates short and long haul flight times slightly differently, which may be the cause of the slight discrepancies between this figure and the previous one.)

Of course, these two data sets are “cheating” since they come from a model which already presupposes what the radius of the Earth is. But one can input real world flight times between these four cities instead of the above idealised data. Here one runs into the issue that the flight time from to is not necessarily the same as that from to due to such factors as windspeed. For instance, I looked up the online flight time from Tokyo to Dubai to be 11 hours and 10 minutes, whereas the online flight time from Dubai to Tokyo was 9 hours and 50 minutes. The simplest thing to do here is take an arithmetic mean of the two times as a preliminary estimate for the flight time without windspeed factors, thus for instance the Tokyo-Dubai flight time would now be 10 hours and 30 minutes, and more generally

This data is not too far off from the online calculator data, but it does distort the graph slightly (taking as before):

Now one gets estimates for the radius of the Earth that are off by about a factor of from the truth, although the round Earth model still is twice as accurate as the flat Earth model .

Given that windspeed should additively affect flight velocity rather than flight time, and the two are inversely proportional to each other, it is more natural to take a harmonic mean rather than an arithmetic mean. This gives the slightly different values

but one still gets essentially the same plot:

So the inaccuracies are presumably coming from some other source. (Note for instance that the true flight time from Tokyo to Dubai is about greater than the calculator predicts, while the flight time from LA to Dubai is about less; these sorts of errors seem to pile up in this calculation.) Nevertheless, it does seem that flight time data is (barely) enough to establish the roundness of the Earth and obtain a somewhat ballpark estimate for its radius. (I assume that the fit would be better if one could include some Southern Hemisphere cities such as Sydney or Santiago, but I was not able to find a good quadruple of widely spaced cities on both hemispheres for which there were direct flights between all six pairs.)

### Voting tactically in the EU elections

This post is addressed at anyone who is voting in Great Britain in the forthcoming elections to the European Parliament and whose principal aim is to maximize the number of MEPs from Remain-supporting parties, where those are deemed to be the Liberal Democrats, the Greens, Change UK, Plaid Cymru and the Scottish National Party. If you have other priorities, then the general principles laid out here may be helpful, but the examples of how to apply them will not necessarily be appropriate to your particular concerns.

What is the voting system?The system used is called the d’Hondt system. The country is divided into a number of regions, and from each region several MEPs will be elected. You get one vote, and it is for a party rather than a single candidate. Once the votes are in, there are a couple of ways of thinking about how they translate into results. One that I like is to imagine that the parties have the option of assigning their votes to their candidates as they wish, and once the assignments have been made, the candidates with the most votes get seats, where is the number of MEPs representing the given region.

For example, if there are three parties for four places, and their vote shares are 50%, 30% and 20%, then the first party will give 25% to two candidates and both will be elected. If the second party tries a similar trick, it will only get one candidate through because the 20% that goes to the third party is greater than the 15% going to the two candidates from the second party. So the result is two candidates for the first party, one for the second and one for the third.

If the vote shares had been 60%, 25% and 15%, then the first party could afford to split three ways and the result would be three seats for the first party and one for the second.

The way this is sometimes presented is as follows. Let’s go back to the first case. We take the three percentages, and for each one we write down the results of dividing it by 1, 2, 3, etc. That gives us the (approximate) numbers

50%, 25%, 17%, 13%, 10%, …

30%, 15%, 10%, 8%, 6%, …

20%, 10%, 7%, 5%, 3%, …

Looking at those numbers, we see that the biggest four are 50%, 25% from the top row, 30% from the second row, and 20% from the third row. So the first party gets two seats, the second party one and the third party one.

How does this affect how I should vote?The answer to this question depends in peculiar ways on the polling in your region. Let’s take my region, Eastern England, as an example. This region gets seven MEPs, and the latest polls show these kinds of percentages.

Brexit Party 40%

Liberal Democrats 17%

Labour 15%

Greens 10%

Conservatives 9%

Change UK 4%

UKIP 2%

If the percentages stay as they are, then the threshold for an MEP is 10%. The Brexit Party gets four MEPs, and the Lib Dems, Labour and the Greens one each. But because the Brexit Party, the Greens and the Conservatives are all close to the 10% threshold, small swings can make a difference to which two out of the fourth Brexit Party candidate, the Green candidate, or the Conservative candidate gets left out. On the other hand, it would take a much bigger swing — of 3% or so — to give the second Lib Dem candidate a chance of being elected. So if your main concern is to maximize the number of Remain-supporting MEPs, you should support the Greens.

Yes, but what if everybody were to do that?In principle that is an annoying problem with the d’Hondt system. But don’t worry — it just isn’t going to happen. Systematic tactical voting is at best a marginal phenomenon, but fortunately in this region a marginal phenomenon may be all it takes to make sure that the Green candidate gets elected.

Aren’t you being defeatist? What about trying to get two Lib Dems and one Green through?This might conceivably be possible, but it would be difficult, and a risky strategy, since going for that could lead to just one Remain-supporting MEP. One possibility would be for Remain-leaning Labour voters to say to themselves “Well, we’re basically guaranteed an MEP, and I’d much prefer a Remain MEP to either the Conservatives or the Brexit Party, so I’ll vote Green or Lib Dem instead.” If that started showing up in polls, then one would be able to do a better risk assessment. But for now it looks better to make sure that the Green candidate gets through.

I’m not from the Eastern region. Where can I find out how to vote in my region?There is a website called remainvoter.com that has done the analysis. The reason I am writing this post is that I have seen online that a lot of people are highly sceptical about their conclusions, so I wanted to explain the theory behind them (as far as I can guess it) so that you don’t have to take what they say on trust and can do the calculations for yourself.

Just to check, I’ll look at another region and see whether I end up with a recommendation that agrees with that of remainvoter.com.

In the South West, there are six MEPs. A recent poll shows the following percentages.

Brexit Party 42%

Lib Dem 20%

Green Party 12%

Conservatives 9%

Labour 8%

Change UK 4%

UKIP 3%

Dividing the Brexit Party vote by 3 gives 14% and dividing the Lib Dem vote by 2 gives 10%. So as things stand there would be three Brexit Party MEPs, two Lib Dem MEPs and one Green Party MEP.

This is a bit close for comfort, but the second Lib Dem candidate is in a more precarious position than the Green Party candidate, given that the Conservative candidate is on 9%. So it would make sense for a bit of Green Party support to transfer to the Lib Dems in order to be sure that the three Remain-supporting candidates that look like being elected in the south west really are.

Interestingly, remainvoter.com recommend supporting the Greens on the grounds that one Lib Dem MEP is bound to be elected. I’m not sure I understand this, since it seems very unlikely that the Lib Dems and the Greens won’t get at least two seats between them, so they might as well aim for three. Perhaps someone can enlighten me on this point. It could be that remainvoter.com is looking at different polls from the ones I’m looking at.

I’m slightly perturbed by that so I’ll pick another region and try the same exercise. Perhaps London would be a good one. Here we have the following percentages (plus a couple of smaller ones that won’t affect anything).

Liberal Democrats 24% (12%, 8%)

Brexit Party 21% (10.5%, 7%)

Labour 19% (9.5%, 6.3%)

Green Party 14% (7%)

Conservatives 10%

Change UK 6%

London has eight MEPs. Here I find it convenient to use the algorithm of dividing by 1,2,3 etc., which explains the percentages I’ve added in brackets. Taking the eight largest numbers we see that the current threshold to get an MEP is at 9.5%, so the Lib Dems get two, the Brexit party two, Labour two and the Greens and Conservatives one each.

Here it doesn’t look obvious how to vote tactically. Clearly not Green, since the Greens are squarely in the middle of the range between the threshold and twice the threshold. Probably not Lib Dem either (unless things change quite a bit) since they’re unlikely to go up as far as 28.5%. But getting Change UK up to 9.5% also looks pretty hard to me. Perhaps the least difficult of these difficult options is for the Green Party to donate about 3% of the vote and the Lib Dems another 2% to Change UK, which would allow them to overtake Labour. But I don’t see it happening.

And now to check my answer, so to speak. And it does indeed agree with the remainvoter.com recommendation. This looks to me like a case where if tactical voting were to be widely adopted, then it might just work to get another MEP, but if it were that widely adopted, one might have to start worrying about not overshooting and accidentally losing one of the other Remain MEPs. But that’s not likely to happen, and in fact I’d predict that in London Change UK will not get an MEP because not enough people will follow remainvoter.com’s recommendation.

This all seems horribly complicated. What should I do?If you don’t want to bother to think about it, then just go to remainvoter.com and follow their recommendation. If you do want to think, then follow these simple (for a typical reader of this blog anyway) instructions.

1. Google polls for your region. (For example, you can scroll down to near the bottom of this page to find one set of polls.)

2. Find out how many MEPs your region gets. Let that number be .

3. For each percentage, divide it by 1, 2, 3 etc. until you reach a number that clearly won’t be in the top .

4. See what percentage, out of all those numbers, is the th largest.

5. Vote for a Remain party that is associated with a number that is close to the threshold if there is also a Brexit-supporting (or Brexit-fence-sitting) party with a number close to the threshold.

One can refine point 5 as follows, to cover the case when more than one Remain-supporting party has a number near the threshold. Suppose, for the sake of example, that the Brexit party is polling at 32%, the Lib Dems at 22%, the Greens at 11%, Labour at 18% and the Conservatives at 12%, and others 5%, in a region that gets five MEPs. Then carrying out step 3, we get

Brexit 32, 16, 10.6

Lib Dems 22, 11

Greens 11

Conservatives 12

Labour 18, 9

So as things stand the Brexit Party gets two MEPs, the Lib Dems one, Labour one and the Conservatives one. If you’re a Remain supporter who wants to vote tactically, then you’ll want to push one of the Lib Dems and the Greens over 12% to defeat the Conservative candidate. To do that, you’ll need either to increase the Green vote from 11% to 12% or to increase the Lib Dem vote from 22% to 24%. The latter is probably harder, so you should probably support the Greens.

A final wordI’m not writing this as an expert, so don’t assume that everything I’ve written is correct, especially given that I came to a different conclusion from remainvoter.com in the South West. If you think I’ve slipped up, then please let me know in the comments, and if I agree with you I’ll make changes. But bear in mind the premise with which I started. Of course there may well be reasons for not voting tactically, such as caring about issues other than Brexit. But this post is about what to do if Brexit is your overriding concern. And one obvious last point: PLEASE ACTUALLY BOTHER TO VOTE! Just the percentage of people voting for Remain-supporting parties will have an impact, even if Farage gets more MEPs.