# Matematički blogovi

### IPAM program in quantitative linear algebra, Mar 19-Jun 15 2018

Alice Guionnet, Assaf Naor, Gilles Pisier, Sorin Popa, Dimitri Shylakhtenko, and I are organising a three month program here at the Institute for Pure and Applied Mathematics (IPAM) on the topic of Quantitative Linear Algebra. The purpose of this program is to bring together mathematicians and computer scientists (both junior and senior) working in various quantitative aspects of linear operators, particularly in large finite dimension. Such aspects include, but are not restricted to discrepancy theory, spectral graph theory, random matrices, geometric group theory, ergodic theory, von Neumann algebras, as well as specific research directions such as the Kadison-Singer problem, the Connes embedding conjecture and the Grothendieck inequality. There will be several workshops and tutorials during the program (for instance I will be giving a series of introductory lectures on random matrix theory).

While we already have several confirmed participants, we are still accepting applications for this program until Dec 4; details of the application process may be found at this page.

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### Yau’s conjecture of abundance of minimal hypersurfaces is generically true (in low dimensions)

Last week, my friends Fernando and André uploaded to the arXiv their remarkable paper “Denseness of minimal hypersurfaces for generic metrics” joint with Kei Irie (and, in fact, Fernando sent me a copy of this article about one day before arXiv’s public announcement).

The motivation for the work of Irie–Marques–Neves is a famous conjecture of Yau on the abundance of minimal surfaces.

More precisely, Yau conjectured in 1982 that a closed Riemannian -manifold contains infinitely many (smooth) closed immersed minimal surfaces. Despite all the activity around this conjecture, the existence of infinitely many embedded minimal hypersurfaces in manifolds of positive Ricci curvature of low dimensions was only established very recently by Fernando and André.

In their remarkable paper, Irie–Marques–Neves show that Yau’s conjecture is *generically* true in low dimensions by establishing the following *stronger* statement:

**Theorem 1** *Let be a closed manifold of dimension . Then, a generic Riemannian metric on has a lot of minimal hypersurfaces: the union of all of its closed (smooth) embedded minimal hypersurfaces is a dense subset of .*

**Remark 1** *The hypothesis of low dimensionality is related to the fact that area-minimizing minimal hypersurfaces in dimensions might exhibit non-trivial singular sets (as it was famously proved by Bombieri–De Giorgi–Guisti), but such a phenomenon does not occur in low dimensions for “min-max” minimal hypersurfaces thanks to the regularity theories of Almgren, Pitts and Schoen–Simon.*

The remainder of this post is dedicated to the proof of this theorem and, as usual, all eventual errors/mistakes in what follows are my entire responsibility.

**1. Description of the key ideas**

Let be a closed manifold and be the space of Riemannian metrics on .

Given an open subset , let be the subset of Riemannian metrics possessing a *non-degenerate*closed (smooth) embedded minimal hypersurface passing through . (Here, *non-degenerate* means that all Jacobi fields are trivial.)

It is possible to check that a non-degenerate closed embedded minimal hypersurface in is *persistent*: more concretely, one can use the definition of non-degeneracy and the inverse function theorem to obtain that any Riemannian metric close to possesses an unique closed embedded minimal hypersurface nearby . In particular, every is *open*.

Also, let us observe (for later use) that the non-degeneracy condition is not difficult to obtain:

**Proposition 2** *Let be a closed (smooth) embedded minimal hypersurface in the Riemannian manifold . Then, we can perform conformal perturbations to find a sequence of metrics converging to (in -topology) such that is a non-degenerate minimal hypersurface of for all sufficiently large .*

*Proof:* This statement is Proposition 2.3 in Irie–Marques–Neves paper and its proof goes along the following lines.

Fix a bump function supported in a small neighborhood of and coinciding with the square of the distance function nearby .

The metrics are conformal to , and they converge to in the -topology. Furthermore, the features of the distance function imply that is a minimal hypersurface of such that the Jacobi operator acting on normal vector fields verify

for all . In particular, the spectrum of is derived from the spectrum of by translation by . Therefore, doesn’t belong to the spectrum of for all surfficiently large, and, hence, is a non-degenerate minimal hypersurface of for all large .

Coming back to Theorem 1, we affirm that our task is reduced to prove the following statement:

**Theorem 3** *Let be a closed manifold of dimension . Then, for any open subset , one has that is dense in .*

In fact, *assuming* Theorem 3, we can easily deduce Theorem 1: if is a countable basis of open subsets of , then Theorem 3 ensures that is a countable intersection of open and dense subsets of the Baire space ; in other terms, is a residual / generic subset of such that any satisfies the conclusions of Theorem 1 (thanks to the definition of and our choice of ).

**Remark 2** *Note that, since is a Baire space, it follows from Baire category theorem that is a dense subset of .*

Let us now explain the proof of Theorem 3. Given a neighborhood of a smooth Riemannian metric on a closed manifold , and an open subset , our goal is to show that

For this sake, we apply White’s bumpy metric theorem asserting that we can find such that all closed (smooth) immersed minimal hypersurfaces in are non-degenerate.

If , then we are done. If , then all closed (smooth) embedded minimal hypersurfaces in avoid . In this case, we can *naively* describe the idea of Irie–Marques–Neves to perturb to get as follows:

- we perturb
*only*in to obtain whose volume is*strictly*larger than the volume ; - by the so-called
*Weyl law for the volume spectrum*(conjectured by Gromov and recently proved by Liokumovich–Marques–Neves), the –*widths*of are strictly larger than those of ; - since -widths “count” the minimal hypersurfaces, the previous item implies that
*new*minimal hypersurfaces in were produced; - because coincides with outside , the minimal hypersurfaces of avoiding are the exactly same of ; thus, the new minimal hypersurfaces in mentioned above must intersect , i.e., .

In the sequel, we will explain how a slight *variant* of this scheme completes the proof of Theorem 3.

**2. Increasing the volume of Riemannian metrics**

Let as above. Take a non-negative smooth bump function supported in such that for some .

Consider the family of conformal deformations of . Note that for all .

From now on, we fix once and for all such that for all .

**3. Weyl law for the volume spectrum**

Roughly speaking, the –*width* of a Riemannian manifold is the following min-max quantity. We consider the space of closed hypersurfaces of , and –*sweepouts* of , i.e., is a continuous map from the -dimensional real projective space to which is “homologically non-trivial” and, *a fortiori*, is *not* a constant map.

**Remark 3** *Intuitively, a -sweepout is a non-trivial way of filling with -parameter family of hypersurfaces (which is “similar” to the way the -parameter family of curves fills the round -sphere ).*

If we denote by the set of -sweepouts such that no concentration of mass occur (i.e., ), then the -width is morally given by

**Remark 4** *Formally speaking, the definition of -width involves more general objects than the ones presented above: we construct by replacing hypersurfaces by certain -dimensional flat chains modulo two, we allow arbitrary simplicial complexes in place of , etc.: see Irie–Marques–Neves’ paper for more details and/or references.*

The -width varies *continuously* with (cf. Lemma 2.1 in Irie–Marques–Neves’ paper). Moreover, it “counts” minimal hypersurfaces (cf. Proposition 2.2 in Irie–Marques–Neves’ paper): if has dimension , then, for each , there is a finite collection of mutually disjoint, closed, smooth, embedded, minimal hypersurfaces in with (stability) indices such that

for some integers . (Here, the stability index is the quantity of negative eigenvalues of the Jacobi operator.)

Furthermore, the asymptotic behavior of is described by Weyl’s law for the volume spectrum (conjectured by Gromov and confirmed by Liokumovich–Marques–Neves): for some *universal* constant , one has

In particular, coming back to the context of Section 2, Weyl’s law for the volume spectrum and the fact that has volume strictly larger than mean that we can select such that the -width is *strictly* larger than the -width of , i.e.,

**4. New minimal hypersurfaces intersecting **

We affirm that there exists such that possesses a closed (smooth) embedded minimal hypersurface passing through .

Otherwise, for each , all closed (smooth) embedded minimal hypersurface in would avoid . Since coincides with outside (by construction), the “counting property of -widths” in equation (1) would imply that

for all .

On the other hand, the fact that is bumpy (in the sense of White’s theorem) permits to conclude that the set is *countable*: indeed, a recent theorem of Sharp implies that the collection of connected, closed, smooth embedded minimal hypersurfaces in with bounded index and volume is finite, so that is countable.

Since the -width depends continuously on the metric, the countability of implies that the function is constant on . In particular, we would have

a contradiction with (2). So, our claim is proved.

At this point, the argument is basically complete: the metric has a closed (smooth) embedded minimal hypersurface passing through ; by Proposition 2, we can perturb (if necessary) in order to get a metric such that is a non-degenerate closed (smooth) embedded minimal hypersurface in , that is, , as desired.

This proves Theorem 3 and, consequently, Theorem 1.

### Serge Cantat’s Bourbaki seminar talk 2017

About one week ago, Serge Cantat gave a beautiful talk in Bourbaki seminar about the recent works of Brown–Fisher–Hurtado on Zimmer’s program. The video of this talk and the corresponding lecture notes are available here and here.

In this post, I will transcript my notes of this talk: as usual, all errors/mistakes are my sole responsibility.

**1. Introduction**

General philosophy behind Zimmer’s program: given a compact manifold (say, the -dimensional sphere), we would like to describe the geometrical and algebraic properties of groups of finite type acting faithfully on ; conversely, given our favorite group of finite type, we want to know the class of compact manifolds on which acts faithfully; in this context, Zimmer’s program proposes some answers to these problems when is a lattice in a Lie group.

More precisely, let be a connected Lie group with finite center whose Lie algebra is semi-simple, and let be a connected maximal split torus of . The dimension of , or equivalently, the dimension of the Lie algebra of , is the so-called *real rank* of , and it is denoted by . Let be a lattice of , i.e., a discrete subgroup such that the quotient has finite Haar measure.

For the sake of concreteness, today we will deal exclusively with the prototypical case of and is subgroup of diagonal matrices in with positive entries.

In this setting, Zimmer’s program offers restrictions on the dimension of compact manifolds admitting non-trivial actions of by smooth diffeomorphisms. In this direction, Aaron Brown, David Fisher and Sebastian Hurtado showed here that

**Theorem 1** *Let be a connected compact manifold. Suppose that the lattice of is uniform (i.e., is compact).*If there exists a homomorphism with infinite image, then

As we are going to see below, the proof of this theorem is a beautiful blend of ideas from geometric group theory and dynamical systems.

Before describing the arguments of Brown–Fisher–Hurtado, let us make a few comments of the statement of their theorem.

**Remark 1** *The assumption of compactness of is important: indeed, any countable group acts faithfully by biholomorphisms of a connected non-compact Riemann surface (see the footnote 1 of Cantat’s text for a short proof of this fact).*

**Remark 2** *The hypothesis of uniformity of is technical: there is some hope to treat non-uniform lattices and, in fact, Brown–Fisher–Hurtado managed to recently extend their result to the case of .*

**Remark 3** *The conclusion is optimal: (and, a fortiori, any lattice of ) acts on the real projective space by projective linear transformations. On the other hand, if one changes , then the inequality can be improved: for example, Brown–Fisher–Hurtado proves that when is the symplectic group of real rank .*

**Remark 4** *Concerning the regularity of the elements of , even though one expects similar statements for actions by homeomorphisms, Brown–Fisher–Hurtado deals only with -diffeomorphisms because they need to employ the so-called Pesin theory of non-uniform hyperbolicity.**Nevertheless, we shall assume that in the sequel for a technical reason explained later.*

**Remark 5** *This theorem is obvious when : indeed, a compact manifold whose group of diffeomorphisms is infinite has dimension .**Hence, we can (and do) assume without loss of generality that in what follows.*

**Remark 6** *The statement of Brown–Fisher–Hurtado theorem is comparable to Margulis super-rigidity theorem providing a control on the dimension of linear representations of .*

**2. Preliminaries**

Fix a Riemannian metric on . Denote by a finite set of generators of . For the sake of convenience, we suppose that contains the identity element and is symmetric (i.e., if and only if ).

The length of a word on the letters of the alphabet is denoted by : in other words, is the distance between and in the Cayley graph of .

The ball of radius is

We say that an action is *feeble* whenever if , there exists such that

for all . (Here, stands for the derivative, and the norm is measured with respect to the Riemannian metric fixed above.) Also, we say that an action is *vigorous* if it is not feeble.

The proof of Theorem 1 is naturally divided into two regimes depending on whether is feeble or vigorous.

**3. Feeble actions**

Our goal in this section is to show that if is *feeble*, then there exists a -invariant Riemannian metric on .

Before proving this claim, let us see why it allows to establish Theorem 1 for feeble actions. The claim implies that is a subgroup of isometries of . Since is compact, Myers–Steenrod theorem says that its group of isometries is a compact Lie group. This permits to apply Margulis super-rigidity theorem and the classical theory of compact Lie groups to get the desired inequality .

Let us now prove the claim.

The first ingredient is a lemma of Fisher–Margulis ensuring that a feeble action is “feeble to all orders”, i.e., for all and , there exists such that

for all .

The second ingredient is provided by the so-called *reinforced property (T)* of Lafforgue. In a nutshell, this property says the following. Given a Hilbert space, denote by the group of continuous linear operators. Let be a parameter. We say that a representation is –*moderate* if there exists such that

for all . Given such a representation , we denote by the set of -invariant vectors. The next statement described the reinforced property (T).

**Theorem 2 (Lafforgue, de Laat–de la Salle)** *Let be a uniform lattice of of . Then, there exist*

- (1)
- (2) probability measures on supported on

*such that for all -moderate , one has a projection with*

**Remark 7** *de la Salle is currently working on extending this result to non-uniform lattices.*

We want to explore this theorem to produce the desired invariant Riemannian metric in the claim.

Since any Riemannian metric is a section of , let us consider the action of on induced by .

Denote by the Hilbert space of sections of whose first derivatives are (i.e., is a Sobolev space of type ).

**Remark 8** *Sobolev embedding theorem implies that an element of is when .*

Observe that the action of on gives a representation

Take a small parameter, where and are the quantities provided the reinforced property (T). By Theorem 2, we have a projection such that

In other words, is a -invariant, -section of which is the limit of the Riemannian metrics

In particular, is non-negative definite. At this point, our task is reduced to prove that is a Riemannian metric, i.e., for all . For this sake, note that if , then the inequality above would give

for all . On the other hand, the action of is feeble of all orders, so that

Since , we get a contradiction unless , i.e., .

This completes the proof of Theorem 1 for feeble actions.

**Remark 9** *We used that here: indeed, we took sufficiently large to apply Sobolev embedding theorem in order to obtain a -smooth object and we exploited the fact that is feeble of order to conclude that is a Riemannian metric.**In the case of actions , one replaces the Hilbert spaces by Banach spaces , and one employs the version of the reinforced property (T) for Banach spaces.*

**4. Vigorous actions**

In this section, we assume that is vigorous.

Roughly speaking, we are going to treat the case of vigorous actions by exploring the tension between the vigour of the action (creating non-zero Lyapunov exponents) and Zimmer’s super-rigidity theorem for cocycles (saying that the Lyapunov exponents of the action with respect to any invariant probability measure vanish when ).

Logically, the problem with this strategy is the fact that is not amenable, so that the existence of invariant probability measures (required by Zimmer’s super-rigidity theorem) is far from being automatic. In particular, this partly explains why the first versions of Zimmer’s program dealt exclusively with actions of by *volume-preserving* diffeomorphisms of . Also, even if we disposed of invariant probability measures, their supports could be very “thin”, so that their generic points would not “feel” the vigour of the action (and hence no contradiction could be derived).

Anyhow, we will discuss how to overcome the difficulties in the previous paragraph: we shall use the vigour of the action in order to construct an invariant probability measure with some positive Lyapunov exponent, so that the desired conclusion will follow from Zimmer’s super-rigidity.

**4.1. Suspensions**

We start by replacing the action of by a `cousin’ action of . More concretely, consider the product space . Note that acts on via

and acts on via

In particular, acts on the space (because the actions of and commute).

Observe that the action of on is a suspension of the action of on with respect to the natural projection .

We denote by is the vertical tangent bundle (i.e., the tangent space to the fibers of ). Let be the restriction of the derivative of to . Given and a -invariant probability measure on , we define

the maximal vertical Lyapunov exponent of with respect to .

**Remark 10** *For each fixed and , the quantity is a continuous function of . Therefore, for each fixed , the Lyapunov exponent is a upper semi-continuous function of .*

Our goal is to exhibit a probability measure on which is -invariant and possessing a positive Lyapunov exponent, i.e.,

for some .

**4.2. -invariant measures**

The first step towards our goal consists in constructing a probability measure on which is -invariant, -ergodic and possessing a positive Lyapunov exponent in the sense that

for some .

For this sake, we recall that a vigorous action has the property that for some and for a sequence , , one has

By Cartan’s decomposition , where is a maximal compact subgroup. Thus, we can write . By compactness, is uniformly bounded for all , so that

where , . (Here we used that and the size of are comparable [because is compact].)

By extracting a subsequence of , we can assume that goes to infinity in a fixed direction with a fixed speed. In particular, this allows us to replace by the iterates of a single element , i.e., we found an element of with

If we take such that , then it is not hard to see that the sequence of probability measures

accumulate into some -invariant probability measure with

Since is amenable and commutes with , we can replace by an -invariant probability measure (by taking averages along F{\o}lner sequences) whose Lyapunov exponent is positive (thanks to the upper semi-continuity property mentioned in Remark 10). Finally, by taking an appropriate ergodic component, we can also assume that is -ergodic.

**4.3. Ratner theory and higher rank groups**

We affirm that the probability measure constructed above can be chosen so that its projection to is

Indeed, the assumption that implies that contains unipotent subgroups commuting with . Since an unipotent subgroup is amenable, we can repeat the argument of the previous subsection (with replaced by such unipotents) to get an *additional* invariance, i.e., we can assume that is invariant under *and* some unipotent subgroup. At this point, Ratner’s theory permits to control the projection and, in particular, to assert that is the Haar measure.

**4.4. Entropy argument**

Let us now show that described above is -invariant on *if* . Note that this completes the proof of Theorem 1 for vigorous actions because the positivity of the Lyapunov exponent contradicts Zimmer’s super-rigidity theorem *unless* .

The vertical Lyapunov exponents of the elements of with respect to define *linear (Lyapunov) forms*

The total number of linear forms (counting multiplicities) is . Since we are assuming that , there exists with

for all .

Let . Recall that the Lyapunov forms associated to the action of on are the trivial form and the roots of . Thus, Pesin entropy formula says that the entropy of the action of on *coincides* with the sum of positive Lyapunov exponent:

(where is the root space of ).

On the other hand, Margulis–Ruelle inequality says that the entropy of action of on is *bounded* by the sum of positive Lyapunov exponents. Since the vertical Lyapunov exponents of vanish, we conclude that

Because projects onto , we also have

In summary, we obtain that Margulis–Ruelle inequality is actually an *equality*. By the invariance theorem of Ledrappier–Young, we derive that is invariant by for all roots with .

By reversing the time (i.e., replacing by ) in the previous argument, we also obtain that is invariant by for all roots with .

Since is the smallest group containing all with (as is a simple Lie group), we conclude the desired -invariance of .

This ends the proof of Theorem 1.

### UCLA Math Undergraduate Merit Scholarship for 2018

In 2010, the UCLA mathematics department launched a scholarship opportunity for entering freshman students with exceptional background and promise in mathematics. We are able to offer one scholarship each year. The UCLA Math Undergraduate Merit Scholarship provides for full tuition, and a room and board allowance for 4 years, contingent on continued high academic performance. In addition, scholarship recipients follow an individualized accelerated program of study, as determined after consultation with UCLA faculty. The program of study leads to a Masters degree in Mathematics in four years.

More information and an application form for the scholarship can be found on the web at:

http://www.math.ucla.edu/ugrad/mums

To be considered for Fall 2018, candidates must apply for the scholarship and also for admission to UCLA on or before November 30, 2017.

Filed under: advertising Tagged: scholarship, UCLA, undergraduate study

### The logarithmically averaged and non-logarithmically averaged Chowla conjectures

Let be the Liouville function, thus is defined to equal when is the product of an even number of primes, and when is the product of an odd number of primes. The Chowla conjecture asserts that has the statistics of a random sign pattern, in the sense that

for all and all distinct natural numbers , where we use the averaging notation

For , this conjecture is equivalent to the prime number theorem (as discussed in this previous blog post), but the conjecture remains open for any .

In recent years, it has been realised that one can make more progress on this conjecture if one works instead with the logarithmically averaged version

of the conjecture, where we use the logarithmic averaging notation

Using the summation by parts (or telescoping series) identity

it is not difficult to show that the Chowla conjecture (1) for a given implies the logarithmically averaged conjecture (2). However, the converse implication is not at all clear. For instance, for , we have already mentioned that the Chowla conjecture

is equivalent to the prime number theorem; but the logarithmically averaged analogue

is significantly easier to show (a proof with the Liouville function replaced by the closely related Möbius function is given in this previous blog post). And indeed, significantly more is now known for the logarithmically averaged Chowla conjecture; in this paper of mine I had proven (2) for , and in this recent paper with Joni Teravainen, we proved the conjecture for all odd (with a different proof also given here).

In view of this emerging consensus that the logarithmically averaged Chowla conjecture was easier than the ordinary Chowla conjecture, it was thus somewhat of a surprise for me to read a recent paper of Gomilko, Kwietniak, and Lemanczyk who (among other things) established the following statement:

**Theorem 1** Assume that the logarithmically averaged Chowla conjecture (2) is true for all . Then there exists a sequence going to infinity such that the Chowla conjecture (1) is true for all along that sequence, that is to say

for all and all distinct .

This implication does not use any special properties of the Liouville function (other than that they are bounded), and in fact proceeds by ergodic theoretic methods, focusing in particular on the ergodic decomposition of invariant measures of a shift into ergodic measures. Ergodic methods have proven remarkably fruitful in understanding these sorts of number theoretic and combinatorial problems, as could already be seen by the ergodic theoretic proof of Szemerédi’s theorem by Furstenberg, and more recently by the work of Frantzikinakis and Host on Sarnak’s conjecture. (My first paper with Teravainen also uses ergodic theory tools.) Indeed, many other results in the subject were first discovered using ergodic theory methods.

On the other hand, many results in this subject that were first proven ergodic theoretically have since been reproven by more combinatorial means; my second paper with Teravainen is an instance of this. As it turns out, one can also prove Theorem 1 by a standard combinatorial (or probabilistic) technique known as the second moment method. In fact, one can prove slightly more:

**Theorem 2** Let be a natural number. Assume that the logarithmically averaged Chowla conjecture (2) is true for . Then there exists a set of natural numbers of logarithmic density (that is, ) such that

for any distinct .

It is not difficult to deduce Theorem 1 from Theorem 2 using a diagonalisation argument. Unfortunately, the known cases of the logarithmically averaged Chowla conjecture ( and odd ) are currently insufficient to use Theorem 2 for any purpose other than to reprove what is already known to be true from the prime number theorem. (Indeed, the even cases of Chowla, in either logarithmically averaged or non-logarithmically averaged forms, seem to be far more powerful than the odd cases; see Remark 1.7 of this paper of myself and Teravainen for a related observation in this direction.)

We now sketch the proof of Theorem 2. For any distinct , we take a large number and consider the limiting the second moment

We can expand this as

If all the are distinct, the hypothesis (2) tells us that the inner averages goes to zero as . The remaining averages are , and there are of these averages. We conclude that

By Markov’s inequality (and (3)), we conclude that for any fixed , there exists a set of upper logarithmic density at least , thus

such that

By deleting at most finitely many elements, we may assume that consists only of elements of size at least (say).

For any , if we let be the union of for , then has logarithmic density . By a diagonalisation argument (using the fact that the set of tuples is countable), we can then find a set of natural numbers of logarithmic density , such that for every , every sufficiently large element of lies in . Thus for every sufficiently large in , one has

for some with . By Cauchy-Schwarz, this implies that

interchanging the sums and using and , this implies that

We conclude on taking to infinity that

as required.

Filed under: expository, math.CO, math.DS, math.NT, math.PR Tagged: Chowla conjecture, second moment method

### The Mihalik-Wieczorek Problem

I already discussed the very interesting mathematics related to the Bohr-Pál Theorem that I learnt about while writing my preprint with W. Sawin on the support of Kloosterman path. This also led us (very) tangentially to what turns out to be an open geometric problem: Mihalik and Wieczorek asked whether there exists a continuous function which sends every interval in to a *convex* subset of , and whose image is *not* contained in an affine line in . In other words: suppose is continuous and satisfies the intermediate value property (in the sense that for any , any point on the segment between and is of the form for some between and ); is the image of contained in some affine line?

The existence of such functions may seem unlikely, but experience shows that being unlikely to exist has rarely stopped functions from actually existing.

The arithmetic relevance of a hypothetical Mihalik-Wieczorek function is that, by adapting Sahakyan’s argument (as discussed before), it seems that we would be able to construct a function in the support of Kloosterman paths that has image containing an open set, answering one of our lingering questions.

However, the question remains apparently open. The best known result (that we know about) is due to Pach and Rogers (1982), and independently Vince and Wilson (1984): there exists , with image not contained in an affine line, such that and are convex for all . (Note that Vince and Wilson conjecture at the end of their paper that a Mihalik-Wieczorek curve does not exist; however their argument is based on a weaker conjecture that this existence would imply, and that statement in turn *is* actually true, as was explicitly stated by Pach and Rogers at the end of their article; see this American Math. Monthly problem.).

### Consoled

Keen-memoried readers will remember the word appearing before on this blog. As one of the happy few who have read *“The Unconsoled”* twice, I applaud with pleasure the honor given to K. Ishiguro! (If my credentials are disputed, let me clearly state that I can answer the question: *“Which spectacular goal scored by a Dutch player during the 1978 World Cup is described in the book?”* — or rather, almost, since the description is ambiguous and could apply to two goals by the same player, during different games; I actually remember watching at least one of them).

### Jean-Christophe Yoccoz mathematical archives

Almost one year ago, Yoccoz family gave me the immense honour of taking care of Jean-Christophe’s mathematical archives.

My general plan is to follow the same steps by Jean-Christophe when he became the responsible for Michel Herman archives, namely, I will make available at this webpage here all unpublished texts after selecting and revising them together with Jean-Christophe’s friends.

So far, the webpage dedicated to Jean-Christophe’s archives contains only an original text (circa 1986), a latex version of this text (typed by Alain Albouy, Alain Chenchiner, and myself), and some lecture notes taken by Alain Chenchiner of a talk by Jean-Christophe on the central configurations for the planar four-body problem.

Nevertheless, I hope that this webpage will be regularly updated in the forthcoming years: indeed, Jean-Christophe’s archives takes all cabinets and some corners of an entire office, and, thus, there is more than enough material to keep his friends occupied for some time.

### The Bohr-Pál Theorem and its friends

In the course of writing our paper on the support of the Kloosterman paths, Will Sawin and I encountered some very beautiful classical questions of Fourier analysis. As discussed in the previous post, we were interested in the following question: given a continuous function , such that (1) we have and (2) the function has purely imaginary Fourier coefficients for , does there exist an increasing homeomorphism , such that , and such that the new function , in addition to the analogue of (1) and (2), also satisfies for ?

After some searching we found the right keywords, or mathematical attractor, for this type of problem. It starts with a short paper of Gyula, alias Julius, alias Jules, Pál, alias Pal, alias Perl, in 1914, who attributes to Fejér the following question : can one reparameterize a continuous periodic function in such a way that the new function has uniformly convergent Fourier series? Pál shows that he can almost do it: his reparameterized function has Fourier series that converges uniformly on , where can be arbitrarily small (but the change of variable depends on ).

This Theorem of Pál becomes the Bohr–Pál Theorem after Harald Bohr’s full answer to Fejér’s question in *Acta Universitatis Szegediensis* in 1935. (Note: I had never looked before at the archive of this particular journal; like most mathematical journals of this period, it is amazing how many names, and even theorems, one can recognize in almost any issue !)

Then comes a beautiful very short proof by Salem in 1944 (it is also the proof found in Zygmund’s book on trigonometric series). All three proofs rely on two results to achieve uniform convergence: one is Riemann’s mapping theorem, and the second is a result of Fejér, according to which the conformal map from the unit disc to the “interior” of a simple closed curve has the property that the boundary Fourier series converges uniformly. In fact, one can certainly write beautiful exercises or exams based on Salem’s proof, for a course in complex analysis that goes as far as Riemann’s Theorem…

The story does not stop here. It is amusing to note that none of Pál, Bohr or Salem feel that it is necessary to point out that they work with *real-valued* periodic functions. But this is essential to the basic structure of their argument (which starts by viewing as the imaginary part of a complex number that describes a simple closed curve in , the trick being to define the real part in a suitable way). Whether the Bohr-Pál Theorem holds for complex-valued functions was an open problem until the late 1970’s (see for instance p. 128 of this 1974 survey of Zalcman about real and complex problems in analysis), when it was proved by Kahane and Katznelson with a completely different approach. They in fact succeeded in proving that one can find a single change of variable that transforms simultaneously all functions in a compact set of the space of (real-valued periodic) continuous functions into functions with uniformly convergent Fourier series; applied to the real and imaginary parts of a complex-valued function, this gives the required extension.

Coming back to our original question, the complex-variable proofs give some information on the Fourier coefficients of the reparameterized function , precisely they imply that

which is encouraging, even if no pointwise estimate follows. But some variants of the question have been studied that get much closer to what we want. They are discussed (among other things) in a nice survey by Olevskii. I find especially fascinating one theorem of Olevskii himself, published in 1981: it is *not* always possible to find a change of variable (of a real-valued periodic continuous function) so that the reparameterized function has an *absolutely convergent* Fourier series. This answers a question of Luzin, and is explained in Section 3 of Olevskii’s survey. Interestingly, this problem is now, of course, *easier* for complex-valued functions, compared to real-valued functions, and it was also solved by Kahane and Katznelson in the -valued case.

The result that turned out to be most useful for our purposes is

due to Sahakyan (whose name is also spelled Saakjan or Saakyan). He proves the existence of reparameterizations of a continuous periodic function satisfying various asymptotic bounds for their Fourier coefficients. The result is here also restricted to real-valued functions, and again for a clear reason: the construction relies crucially on the intermediate value theorem for continuous functions. More precisely, Sahakyan’s idea is to use the *Faber-Schauder* expansions of continuous functions, and to find a change of variable that leads to a function with a “sparse” Faber-Schauder expansion, from which he estimates directly the Fourier coefficients using bounds for those of the Faber-Schauder functions. (I actually didn’t know about Faber-Schauder expansions before; I will also certainly make use of them for analysis or functional analysis exercises later…) This works because the Faber-Schauder coefficients are quite simple: they are of the form

for suitable real numbers . One can now easily imagine how the intermediate value theorem will make it possible to find a change of variable to make such a coefficient vanish.

Using Sahakyan’s main lemma, with a few modifications to take into account the additional symmetry we require, we were able to solve our reparameterization problem for the support of Kloosterman paths, in the case of real-valued functions. The complex-value case, as far as we know, remains open…

### The support of Kloosterman paths

Will Sawin and I just put up on arXiv a preprint that is the natural follow-up to our paper on those most alluring of shapes, the Kloosterman paths.

As the title indicates, we are looking this time at the support of the limiting random Fourier series that arose in that first paper, namely

where is a sequence of independent Sato-Tate-distributed random variables. In a strict sense, this should be a very short paper, since the computation of the support is easily achieved using some basic probability and elementary properties of Fourier series: it is the set of continuous functions such that (1) the value of at is real and belongs to ; (2) the function has purely imaginary Fourier coefficients for ; (3) we have for all .

So why is the paper 26 pages long? The reason is that this support (call it ) is a rather interesting set of functions, and we spend the rest of the paper exploring some of its properties. Most importantly, the support is not *all* functions, so we can play the game of picking our favorite continuous function on (say ) and ask whether or not belongs to .

For instance:

- Fixing a prime , and , invertible modulo , does the Kloosterman path
*itself*belong to the support? Simple computations show that it depends on ! For instance, the path for the Kloosterman sum , shown below, does not belong to the support. (As we observe, it looks like a Shadok, whose mathematical abilities are well-known — sorry, the last link is only in French ; I suggest to every French-aware reader to watch the corresponding episode, since the voice of C. Piéplu achieves the seemingly impossible in making this hilarious text even funnier…)

Kl_2(8,1;19) - On the other hand, the path giving the graph of the Takagi function (namely ) belongs to the support.

Takagi function -
But maybe the most interesting problem from a mathematical point of view is one of pure analysis: when we see a Kloosterman path (such as the one above), we only see its image as a function from to , independently of the parameterization of the path. So we can take any shape in the plane that can be represented as the image of a function satisfying the conditions (1) and (2) above, and ask: is there a
*reparameterization*of that belongs to the suppport? For instance, for the Kloosterman paths themselves (as in (1) above), it is not difficult to find one: instead of following each of the segments making the Kloosterman path in time , one can insert a “pause” of length at the beginning and end of the path, and then divide equally the remaining time for the segments. (The fact that this re-parameterized path, whose image is still the same Kloosterman path, belongs to the support is then an elementary consequence of the Weil bound for Kloosterman sums). -
In general, the question is whether a given has a reparameterization with Fourier coefficients (rather, those of ) are all smaller than . This is an intriguing problem, and looking into it brought us into contact with some very nice classical questions in Fourier analysis, that I discuss in this later post. We only succeeded in proving the existence of a suitable reparameterization for
*real-valued*functions, for reasons explained in the aforementioned later post, and it is an interesting analysis problem whether the result holds for all functions. A positive answer would in particular settle another natural question that we haven’t been able to handle yet:*is there a space-filling curve in the support of the Kloosterman paths*?

All this is great analytic fun. But there are nice arithmetic consequences of our result. By the definition of the support, we know at least that any has the property that, with positive probability, the actual path of the partial sums of the Kloosterman sums will come as close as we want (uniformly on ) to , and this is an arithmetic statement. For instance, simply because the zero function belongs to the support, we deduce that, for a large prime , there is a positive proportion of such that *all* partial sums

for , have modulus .

In other words, there is a non-zero probability that *all* the normalized partial sums of the Kloosterman sums are very small. (It is interesting to note that this is emphetically not true for character sums… the point is that their Fourier expansion involves multiplicative coefficients, so they cannot become smaller than .)

### Tornare a Ventotene

I participated last week to the wonderful Ventotene 2017 Conference, a worthy continuation of Ventotene 2015. Reaching the island required this time even more of the stamina that the conference website recommends, since the weather was rough enough that the faster hydrofoil boat did not run (stranding about 30 of the participants in Formia on Sunday evening), while even the rather bigger one behaved more like a large scale roller-coaster than most people would wish.

After arriving at the island, Monday was still a bit unpleasant (it was much more for those who were unlucky to be exposed when one of the few short but very violent rain showers fell…), but the remaining of the time was beautiful. On the way back, I had to stop in Rome for a night, and tasted the most delicious *ragù bianco di coniglio* that one can imagine.

I’m already looking forward to the next conference…