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Now that Google Plus is closing, the brief announcements that I used to post over there will now be migrated over to this blog. (Some people have suggested other platforms for this also, such as Twitter, but I think for now I can use my existing blog to accommodate these sorts of short posts.)

- The NSF-CBMS regional research conferences are now requesting proposals for the 2020 conference series. (I was the principal lecturer for one of these conferences back in 2005; it was a very intensive experience, but quite enjoyable, and I am quite pleased with the book that resulted from it.)
- The awardees for the Sloan Fellowships for 2019 have now been announced. (I was on the committee for the mathematics awards. For the usual reasons involving the confidentiality of letters of reference and other sensitive information, I will be unfortunately be unable to answer any specific questions about our committee deliberations.)

I have just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows“, submitted to Pure and Applied Functional Analysis. This paper continues my attempts to establish “universality” properties of the Euler equations on Riemannian manifolds , as I conjecture that the freedom to set the metric ought to allow one to “program” such Euler flows to exhibit a wide range of behaviour, and in particular to achieve finite time blowup (if the dimension is sufficiently large, at least).

In coordinates, the Euler equations read

where is the pressure field and is the velocity field, and denotes the Levi-Civita connection with the usual Penrose abstract index notation conventions; we restrict attention here to the case where are smooth and is compact, smooth, orientable, connected, and without boundary. Let’s call an *Euler flow* on (for the time interval ) if it solves the above system of equations for some pressure , and an *incompressible flow* if it just obeys the divergence-free relation . Thus every Euler flow is an incompressible flow, but the converse is certainly not true; for instance the various conservation laws of the Euler equation, such as conservation of energy, will already block most incompressible flows from being an Euler flow, or even being approximated in a reasonably strong topology by such Euler flows.

However, one can ask if an incompressible flow can be *extended* to an Euler flow by adding some additional dimensions to . In my paper, I formalise this by considering warped products of which (as a smooth manifold) are products of with a torus, with a metric given by

for , where are the coordinates of the torus , and are smooth positive coefficients for ; in order to preserve the incompressibility condition, we also require the volume preservation property

though in practice we can quickly dispose of this condition by adding one further “dummy” dimension to the torus . We say that an incompressible flow is *extendible to an Euler flow* if there exists a warped product extending , and an Euler flow on of the form

for some “swirl” fields . The situation here is motivated by the familiar situation of studying axisymmetric Euler flows on , which in cylindrical coordinates take the form

The base component

of this flow is then a flow on the two-dimensional plane which is not quite incompressible (due to the failure of the volume preservation condition (2) in this case) but still satisfies a system of equations (coupled with a passive scalar field that is basically the square of the swirl ) that is reminiscent of the Boussinesq equations.

On a fixed -dimensional manifold , let denote the space of incompressible flows , equipped with the smooth topology (in spacetime), and let denote the space of such flows that are extendible to Euler flows. Our main theorem is

Theorem 1

- (i) (Generic inextendibility) Assume . Then is of the first category in (the countable union of nowhere dense sets in ).
- (ii) (Non-rigidity) Assume (with an arbitrary metric ). Then is somewhere dense in (that is, the closure of has non-empty interior).

More informally, starting with an incompressible flow , one usually cannot extend it to an Euler flow just by extending the manifold, warping the metric, and adding swirl coefficients, even if one is allowed to select the dimension of the extension, as well as the metric and coefficients, arbitrarily. However, many such flows can be *perturbed* to be extendible in such a manner (though different perturbations will require different extensions, in particular the dimension of the extension will not be fixed). Among other things, this means that conservation laws such as energy (or momentum, helicity, or circulation) no longer present an obstruction when one is allowed to perform an extension (basically this is because the swirl components of the extension can exchange energy (or momentum, etc.) with the base components in a basically arbitrary fashion.

These results fall short of my hopes to use the ability to extend the manifold to create universal behaviour in Euler flows, because of the fact that each flow requires a different extension in order to achieve the desired dynamics. Still it does seem to provide a little bit of support to the idea that high-dimensional Euler flows are quite “flexible” in their behaviour, though not completely so due to the generic inextendibility phenomenon. This flexibility reminds me a little bit of the flexibility of weak solutions to equations such as the Euler equations provided by the “-principle” of Gromov and its variants (as discussed in these recent notes), although in this case the flexibility comes from adding additional dimensions, rather than by repeatedly adding high-frequency corrections to the solution.

The proof of part (i) of the theorem basically proceeds by a dimension counting argument (similar to that in the proof of Proposition 9 of these recent lecture notes of mine). Heuristically, the point is that an arbitrary incompressible flow is essentially determined by independent functions of space and time, whereas the warping factors are functions of space only, the pressure field is one function of space and time, and the swirl fields are technically functions of both space and time, but have the same number of degrees of freedom as a function just of space, because they solve an evolution equation. When , this means that there are fewer unknown functions of space and time than prescribed functions of space and time, which is the source of the generic inextendibility. This simple argument breaks down when , but we do not know whether the claim is actually false in this case.

The proof of part (ii) proceeds by direct calculation of the effect of the warping factors and swirl velocities, which effectively create a forcing term (of Boussinesq type) in the first equation of (1) that is a combination of functions of the Eulerian spatial coordinates (coming from the warping factors) and the Lagrangian spatial coordinates (which arise from the swirl velocities, which are passively transported by the flow). In a non-empty open subset of , the combination of these coordinates becomes a non-degenerate set of coordinates for spacetime, and one can then use the Stone-Weierstrass theorem to conclude. The requirement that be topologically a torus is a technical hypothesis in order to avoid topological obstructions such as the hairy ball theorem, but it may be that the hypothesis can be dropped (and it may in fact be true, in the case at least, that is dense in all of , not just in a non-empty open subset).

[*This post is collectively authored by the ICM structure committee, whose membership includes myself, and is listed in full in the post below – T.*]

The International Congress of Mathematicians (ICM) is widely considered to be the premier conference for mathematicians. It is held every four years; for instance, the 2018 ICM was held in Rio de Janeiro, Brazil, and the 2022 ICM is to be held in Saint Petersburg, Russia. The most high-profile event at the ICM is the awarding of the 10 or so prizes of the International Mathematical Union (IMU) such as the Fields Medal, and the lectures by the prize laureates; but there are also approximately twenty plenary lectures from leading experts across all mathematical disciplines, several public lectures of a less technical nature, about 180 more specialised invited lectures divided into about twenty section panels, each corresponding to a mathematical field (or range of fields), as well as various outreach and social activities, exhibits and satellite programs, and meetings of the IMU General Assembly; see for instance the program for the 2018 ICM for a sample schedule. In addition to these official events, the ICM also provides more informal networking opportunities, in particular allowing mathematicians at all stages of career, and from all backgrounds and nationalities, to interact with each other.

For each Congress, a Program Committee (together with subcommittees for each section) is entrusted with the task of selecting who will give the lectures of the ICM (excluding the lectures by prize laureates, which are selected by separate prize committees); they also have decided how to appropriately subdivide the entire field of mathematics into sections. Given the prestigious nature of invitations from the ICM to present a lecture, this has been an important and challenging task, but one for which past Program Committees have managed to fulfill in a largely satisfactory fashion.

Nevertheless, in the last few years there has been substantial discussion regarding ways in which the process for structuring the ICM and inviting lecturers could be further improved, for instance to reflect the fact that the distribution of mathematics across various fields has evolved over time. At the 2018 ICM General Assembly meeting in Rio de Janeiro, a resolution was adopted to create a new Structure Committee to take on some of the responsibilities previously delegated to the Program Committee, focusing specifically on the structure of the scientific program. On the other hand, the Structure Committee is *not* involved with the format for prize lectures, the selection of prize laureates, or the selection of plenary and sectional lecturers; these tasks are instead the responsibilities of other committees (the local Organizing Committee, the prize committees, and the Program Committee respectively).

The first Structure Committee was constituted on 1 Jan 2019, with the following members:

- Terence Tao [Chair from 15 Feb, 2019]
- Carlos Kenig [IMU President (from 1 Jan 2019),
*ex officio*] - Nalini Anantharaman
- Alexei Borodin
- Annalisa Buffa
- Irene Fonseca
- János Kollár
- Laci Lovász [Chair until 15 Feb, 2019]
- Terry Lyons
- Stephane Mallat
- Hiraku Nakajima
- Éva Tardos
- Peter Teichner
- Akshay Venkatesh
- Anna Wienhard

As one of our first actions, we on the committee are using this blog post to solicit input from the mathematical community regarding the topics within our remit. Among the specific questions (in no particular order) for which we seek comments are the following:

- Are there suggestions to change the format of the ICM that would increase its value to the mathematical community?
- Are there suggestions to change the format of the ICM that would encourage greater participation and interest in attending, particularly with regards to junior researchers and mathematicians from developing countries?
- What is the correct balance between research and exposition in the lectures? For instance, how strongly should one emphasize the importance of good exposition when selecting plenary and sectional speakers? Should there be “Bourbaki style” expository talks presenting work not necessarily authored by the speaker?
- Is the balance between plenary talks, sectional talks, and public talks at an optimal level? There is only a finite amount of space in the calendar, so any increase in the number or length of one of these types of talks will come at the expense of another.
- The ICM is generally perceived to be more important to pure mathematics than to applied mathematics. In what ways can the ICM be made more relevant and attractive to applied mathematicians, or should one not try to do so?
- Are there structural barriers that cause certain areas or styles of mathematics (such as applied or interdisciplinary mathematics) or certain groups of mathematicians to be under-represented at the ICM? What, if anything, can be done to mitigate these barriers?

Of course, we do not expect these complex and difficult questions to be resolved within this blog post, and debating these and other issues would likely be a major component of our internal committee discussions. Nevertheless, we would value constructive comments towards the above questions (or on other topics within the scope of our committee) to help inform these subsequent discussions. We therefore welcome and invite such commentary, either as responses to this blog post, or sent privately to one of the members of our committee. We would also be interested in having readers share their personal experiences at past congresses, and how it compares with other major conferences of this type. (But in order to keep the discussion focused and constructive, we request that comments here refrain from discussing topics that are out of the scope of this committee, such as suggesting specific potential speakers for the next congress, which is a task instead for the 2022 ICM Program Committee.)

While talking mathematics with a postdoc here at UCLA (March Boedihardjo) we came across the following matrix problem which we managed to solve, but the proof was cute and the process of discovering it was fun, so I thought I would present the problem here as a puzzle without revealing the solution for now.

The problem involves word maps on a matrix group, which for sake of discussion we will take to be the special orthogonal group of real matrices (one of the smallest matrix groups that contains a copy of the free group, which incidentally is the key observation powering the Banach-Tarski paradox). Given any abstract word of two generators and their inverses (i.e., an element of the free group ), one can define the word map simply by substituting a pair of matrices in into these generators. For instance, if one has the word , then the corresponding word map is given by

for . Because contains a copy of the free group, we see the word map is non-trivial (not equal to the identity) if and only if the word itself is nontrivial.

Anyway, here is the problem:

Problem.Does there exist a sequence of non-trivial word maps that converge uniformly to the identity map?

To put it another way, given any , does there exist a non-trivial word such that for all , where denotes (say) the operator norm, and denotes the identity matrix in ?

As I said, I don’t want to spoil the fun of working out this problem, so I will leave it as a challenge. Readers are welcome to share their thoughts, partial solutions, or full solutions in the comments below.

This is the eleventh research thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.

There are currently two strands of activity. One is writing up the paper describing the combination of theoretical and numerical results needed to obtain the new bound . The latest version of the writeup may be found here, in this directory. The theoretical side of things have mostly been written up; the main remaining tasks to do right now are

- giving a more detailed description and illustration of the two major numerical verifications, namely the barrier verification that establishes a zero-free region for for , and the Dirichlet series bound that establishes a zero-free region for ; and
- giving more detail on the conditional results assuming more numerical verification of RH.

Meanwhile, several of us have been exploring the behaviour of the zeroes of for negative ; this does not directly lead to any new progress on bounding (though there is a good chance that it may simplify the proof of ), but there have been some interesting numerical phenomena uncovered, as summarised in this set of slides. One phenomenon is that for large negative , many of the complex zeroes begin to organise themselves near the curves

(An example of the agreement between the zeroes and these curves may be found here.) We now have a (heuristic) theoretical explanation for this; we should have an approximation

in this region (where are defined in equations (11), (15), (17) of the writeup, and the above curves arise from (an approximation of) those locations where two adjacent terms , in this series have equal magnitude (with the other terms being of lower order).

However, we only have a partial explanation at present of the interesting behaviour of the real zeroes at negative t, for instance the surviving zeroes at extremely negative values of appear to lie on the curve where the quantity is close to a half-integer, where

The remaining zeroes exhibit a pattern in coordinates that is approximately 1-periodic in , where

A plot of the zeroes in these coordinates (somewhat truncated due to the numerical range) may be found here.

We do not yet have a total explanation of the phenomena seen in this picture. It appears that we have an approximation

where is the non-zero multiplier

and

The derivation of this formula may be found in this wiki page. However our initial attempts to simplify the above approximation further have proven to be somewhat inaccurate numerically (in particular giving an incorrect prediction for the location of zeroes, as seen in this picture). We are in the process of using numerics to try to resolve the discrepancies (see this page for some code and discussion).

These lecture notes are a continuation of the 254A lecture notes from the previous quarter.

We consider the Euler equations for incompressible fluid flow on a Euclidean space ; we will label as the “Eulerian space” (or “Euclidean space”, or “physical space”) to distinguish it from the “Lagrangian space” (or “labels space”) that we will introduce shortly (but the reader is free to also ignore the or subscripts if he or she wishes). Elements of Eulerian space will be referred to by symbols such as , we use to denote Lebesgue measure on and we will use for the coordinates of , and use indices such as to index these coordinates (with the usual summation conventions), for instance denotes partial differentiation along the coordinate. (We use superscripts for coordinates instead of subscripts to be compatible with some differential geometry notation that we will use shortly; in particular, when using the summation notation, we will now be matching subscripts with superscripts for the pair of indices being summed.)

In Eulerian coordinates, the Euler equations read

where is the velocity field and is the pressure field. These are functions of time and on the spatial location variable . We will refer to the coordinates as Eulerian coordinates. However, if one reviews the physical derivation of the Euler equations from 254A Notes 0, before one takes the continuum limit, the fundamental unknowns were not the velocity field or the pressure field , but rather the trajectories , which can be thought of as a single function from the coordinates (where is a time and is an element of the label set ) to . The relationship between the trajectories and the velocity field was given by the informal relationship

We will refer to the coordinates as (discrete) *Lagrangian coordinates* for describing the fluid.

In view of this, it is natural to ask whether there is an alternate way to formulate the continuum limit of incompressible inviscid fluids, by using a continuous version of the Lagrangian coordinates, rather than Eulerian coordinates. This is indeed the case. Suppose for instance one has a smooth solution to the Euler equations on a spacetime slab in Eulerian coordinates; assume furthermore that the velocity field is uniformly bounded. We introduce another copy of , which we call *Lagrangian space* or *labels space*; we use symbols such as to refer to elements of this space, to denote Lebesgue measure on , and to refer to the coordinates of . We use indices such as to index these coordinates, thus for instance denotes partial differentiation along the coordinate. We will use summation conventions for both the Eulerian coordinates and the Lagrangian coordinates , with an index being summed if it appears as both a subscript and a superscript in the same term. While and are of course isomorphic, we will try to refrain from identifying them, except perhaps at the initial time in order to fix the initialisation of Lagrangian coordinates.

Given a smooth and bounded velocity field , define a *trajectory map* for this velocity to be any smooth map that obeys the ODE

in view of (2), this describes the trajectory (in ) of a particle labeled by an element of . From the Picard existence theorem and the hypothesis that is smooth and bounded, such a map exists and is unique as long as one specifies the initial location assigned to each label . Traditionally, one chooses the initial condition

for , so that we label each particle by its initial location at time ; we are also free to specify other initial conditions for the trajectory map if we please. Indeed, we have the freedom to “permute” the labels by an arbitrary diffeomorphism: if is a trajectory map, and is any diffeomorphism (a smooth map whose inverse exists and is also smooth), then the map is also a trajectory map, albeit one with different initial conditions .

Despite the popularity of the initial condition (4), we will try to keep conceptually separate the Eulerian space from the Lagrangian space , as they play different physical roles in the interpretation of the fluid; for instance, while the Euclidean metric is an important feature of Eulerian space , it is not a geometrically natural structure to use in Lagrangian space . We have the following more general version of Exercise 8 from 254A Notes 2:

Exercise 1Let be smooth and bounded.

- If is a smooth map, show that there exists a unique smooth trajectory map with initial condition for all .
- Show that if is a diffeomorphism and , then the map is also a diffeomorphism.

Remark 2The first of the Euler equations (1) can now be written in the formwhich can be viewed as a continuous limit of Newton’s first law .

Call a diffeomorphism *(oriented) volume preserving* if one has the equation

for all , where the total differential is the matrix with entries for and , where are the components of . (If one wishes, one can also view as a linear transformation from the tangent space of Lagrangian space at to the tangent space of Eulerian space at .) Equivalently, is orientation preserving and one has a Jacobian-free change of variables formula

for all , which is in turn equivalent to having the same Lebesgue measure as for any measurable set .

The divergence-free condition then can be nicely expressed in terms of volume-preserving properties of the trajectory maps , in a manner which confirms the interpretation of this condition as an incompressibility condition on the fluid:

Lemma 3Let be smooth and bounded, let be a volume-preserving diffeomorphism, and let be the trajectory map. Then the following are equivalent:

- on .
- is volume-preserving for all .

*Proof:* Since is orientation-preserving, we see from continuity that is also orientation-preserving. Suppose that is also volume-preserving, then for any we have the conservation law

for all . Differentiating in time using the chain rule and (3) we conclude that

for all , and hence by change of variables

which by integration by parts gives

for all and , so is divergence-free.

To prove the converse implication, it is convenient to introduce the *labels map* , defined by setting to be the inverse of the diffeomorphism , thus

for all . By the implicit function theorem, is smooth, and by differentiating the above equation in time using (3) we see that

where is the usual material derivative

acting on functions on . If is divergence-free, we have from integration by parts that

for any test function . In particular, for any , we can calculate

and hence

for any . Since is volume-preserving, so is , thus

Thus is volume-preserving, and hence is also.

Exercise 4Let be a continuously differentiable map from the time interval to the general linear group of invertible matrices. Establish Jacobi’s formulaand use this and (6) to give an alternate proof of Lemma 3 that does not involve any integration in space.

Remark 5One can view the use of Lagrangian coordinates as an extension of the method of characteristics. Indeed, from the chain rule we see that for any smooth function of Eulerian spacetime, one hasand hence any transport equation that in Eulerian coordinates takes the form

for smooth functions of Eulerian spacetime is equivalent to the ODE

where are the smooth functions of Lagrangian spacetime defined by

In this set of notes we recall some basic differential geometry notation, particularly with regards to pullbacks and Lie derivatives of differential forms and other tensor fields on manifolds such as and , and explore how the Euler equations look in this notation. Our discussion will be entirely formal in nature; we will assume that all functions have enough smoothness and decay at infinity to justify the relevant calculations. (It is possible to work rigorously in Lagrangian coordinates – see for instance the work of Ebin and Marsden – but we will not do so here.) As a general rule, Lagrangian coordinates tend to be somewhat less convenient to use than Eulerian coordinates for establishing the basic analytic properties of the Euler equations, such as local existence, uniqueness, and continuous dependence on the data; however, they are quite good at clarifying the more algebraic properties of these equations, such as conservation laws and the variational nature of the equations. It may well be that in the future we will be able to use the Lagrangian formalism more effectively on the analytic side of the subject also.

Remark 6One can also write the Navier-Stokes equations in Lagrangian coordinates, but the equations are not expressed in a favourable form in these coordinates, as the Laplacian appearing in the viscosity term becomes replaced with a time-varying Laplace-Beltrami operator. As such, we will not discuss the Lagrangian coordinate formulation of Navier-Stokes here.

Note: this post is not required reading for this course, or for the sequel course in the winter quarter.

In a Notes 2, we reviewed the classical construction of Leray of global weak solutions to the Navier-Stokes equations. We did not quite follow Leray’s original proof, in that the notes relied more heavily on the machinery of Littlewood-Paley projections, which have become increasingly common tools in modern PDE. On the other hand, we did use the same “exploiting compactness to pass to weakly convergent subsequence” strategy that is the standard one in the PDE literature used to construct weak solutions.

As I discussed in a previous post, the manipulation of sequences and their limits is analogous to a “cheap” version of nonstandard analysis in which one uses the Fréchet filter rather than an ultrafilter to construct the nonstandard universe. (The manipulation of generalised functions of Columbeau-type can also be comfortably interpreted within this sort of cheap nonstandard analysis.) Augmenting the manipulation of sequences with the right to pass to subsequences whenever convenient is then analogous to a sort of “lazy” nonstandard analysis, in which the implied ultrafilter is never actually constructed as a “completed object“, but is instead lazily evaluated, in the sense that whenever membership of a given subsequence of the natural numbers in the ultrafilter needs to be determined, one either passes to that subsequence (thus placing it in the ultrafilter) or the complement of the sequence (placing it out of the ultrafilter). This process can be viewed as the initial portion of the transfinite induction that one usually uses to construct ultrafilters (as discussed using a voting metaphor in this post), except that there is generally no need in any given application to perform the induction for any uncountable ordinal (or indeed for most of the countable ordinals also).

On the other hand, it is also possible to work directly in the orthodox framework of nonstandard analysis when constructing weak solutions. This leads to an approach to the subject which is largely equivalent to the usual subsequence-based approach, though there are some minor technical differences (for instance, the subsequence approach occasionally requires one to work with separable function spaces, whereas in the ultrafilter approach the reliance on separability is largely eliminated, particularly if one imposes a strong notion of saturation on the nonstandard universe). The subject acquires a more “algebraic” flavour, as the quintessential analysis operation of taking a limit is replaced with the “standard part” operation, which is an algebra homomorphism. The notion of a sequence is replaced by the distinction between standard and nonstandard objects, and the need to pass to subsequences disappears entirely. Also, the distinction between “bounded sequences” and “convergent sequences” is largely eradicated, particularly when the space that the sequences ranged in enjoys some compactness properties on bounded sets. Also, in this framework, the notorious non-uniqueness features of weak solutions can be “blamed” on the non-uniqueness of the nonstandard extension of the standard universe (as well as on the multiple possible ways to construct nonstandard mollifications of the original standard PDE). However, many of these changes are largely cosmetic; switching from a subsequence-based theory to a nonstandard analysis-based theory does *not* seem to bring one significantly closer for instance to the global regularity problem for Navier-Stokes, but it could have been an alternate path for the historical development and presentation of the subject.

In any case, I would like to present below the fold this nonstandard analysis perspective, quickly translating the relevant components of real analysis, functional analysis, and distributional theory that we need to this perspective, and then use it to re-prove Leray’s theorem on existence of global weak solutions to Navier-Stokes.

Kaisa Matomäki, Maksym Radziwill, and I just uploaded to the arXiv our paper “Fourier uniformity of bounded multiplicative functions in short intervals on average“. This paper is the outcome of our attempts during the MSRI program in analytic number theory last year to attack the local Fourier uniformity conjecture for the Liouville function . This conjecture generalises a landmark result of Matomäki and Radziwill, who show (among other things) that one has the asymptotic

whenever and goes to infinity as . Informally, this says that the Liouville function has small mean for almost all short intervals . The remarkable thing about this theorem is that there is no lower bound on how goes to infinity with ; one can take for instance . This lack of lower bound was crucial when I applied this result (or more precisely, a generalisation of this result to arbitrary non-pretentious bounded multiplicative functions) a few years ago to solve the Erdös discrepancy problem, as well as a logarithmically averaged two-point Chowla conjecture, for instance it implies that

The local Fourier uniformity conjecture asserts the stronger asymptotic

under the same hypotheses on and . As I worked out in a previous paper, this conjecture would imply a logarithmically averaged three-point Chowla conjecture, implying for instance that

This particular bound also follows from some slightly different arguments of Joni Teräväinen and myself, but the implication would also work for other non-pretentious bounded multiplicative functions, whereas the arguments of Joni and myself rely more heavily on the specific properties of the Liouville function (in particular that for all primes ).

There is also a higher order version of the local Fourier uniformity conjecture in which the linear phase is replaced with a polynomial phase such as , or more generally a nilsequence ; as shown in my previous paper, this conjecture implies (and is in fact equivalent to, after logarithmic averaging) a logarithmically averaged version of the full Chowla conjecture (not just the two-point or three-point versions), as well as a logarithmically averaged version of the Sarnak conjecture.

The main result of the current paper is to obtain some cases of the local Fourier uniformity conjecture:

Theorem 1The asymptotic (2) is true when for a fixed .

Previously this was known for by the work of Zhan (who in fact proved the stronger pointwise assertion for in this case). In a previous paper with Kaisa and Maksym, we also proved a weak version

of (2) for any growing arbitrarily slowly with ; this is stronger than (1) (and is in fact proven by a variant of the method) but significantly weaker than (2), because in the latter the worst-case is permitted to depend on the parameter, whereas in (3) must remain independent of .

Unfortunately, the restriction is not strong enough to give applications to Chowla-type conjectures (one would need something more like for this). However, it can still be used to control some sums that had not previously been manageable. For instance, a quick application of the circle method lets one use the above theorem to derive the asymptotic

whenever for a fixed , where is the von Mangoldt function. Amusingly, the seemingly simpler question of establishing the expected asymptotic for

is only known in the range (from the work of Zaccagnini). Thus we have a rare example of a number theory sum that becomes *easier* to control when one inserts a Liouville function!

We now give an informal description of the strategy of proof of the theorem (though for numerous technical reasons, the actual proof deviates in some respects from the description given here). If (2) failed, then for many values of we would have the lower bound

for some frequency . We informally describe this correlation between and by writing

for (informally, one should view this as asserting that “behaves like” a constant multiple of ). For sake of discussion, suppose we have this relationship for *all* , not just *many*.

As mentioned before, the main difficulty here is to understand how varies with . As it turns out, the multiplicativity properties of the Liouville function place a significant constraint on this dependence. Indeed, if we let be a fairly small prime (e.g. of size for some ), and use the identity for the Liouville function to conclude (at least heuristically) from (4) that

for . (In practice, we will have this sort of claim for *many* primes rather than *all* primes , after using tools such as the Turán-Kubilius inequality, but we ignore this distinction for this informal argument.)

Now let and be primes comparable to some fixed range such that

and

on essentially the same range of (two nearby intervals of length ). This suggests that the frequencies and should be close to each other modulo , in particular one should expect the relationship

Comparing this with (5) one is led to the expectation that should depend inversely on in some sense (for instance one can check that

would solve (6) if ; by Taylor expansion, this would correspond to a global approximation of the form ). One now has a problem of an additive combinatorial flavour (or of a “local to global” flavour), namely to leverage the relation (6) to obtain global control on that resembles (7).

A key obstacle in solving (6) efficiently is the fact that one only knows that and are close modulo , rather than close on the real line. One can start resolving this problem by the Chinese remainder theorem, using the fact that we have the freedom to shift (say) by an arbitrary integer. After doing so, one can arrange matters so that one in fact has the relationship

whenever and obey (5). (This may force to become extremely large, on the order of , but this will not concern us.)

Now suppose that we have and primes such that

For every prime , we can find an such that is within of both and . Applying (8) twice we obtain

and

and thus by the triangle inequality we have

for all ; hence by the Chinese remainder theorem

In practice, in the regime that we are considering, the modulus is so huge we can effectively ignore it (in the spirit of the Lefschetz principle); so let us pretend that we in fact have

whenever and obey (9).

Now let be an integer to be chosen later, and suppose we have primes such that the difference

is small but non-zero. If is chosen so that

(where one is somewhat loose about what means) then one can then find real numbers such that

for , with the convention that . We then have

which telescopes to

and thus

and hence

In particular, for each , we expect to be able to write

for some . This quantity can vary with ; but from (10) and a short calculation we see that

whenever obey (9) for some .

Now imagine a “graph” in which the vertices are elements of , and two elements are joined by an edge if (9) holds for some . Because of exponential sum estimates on , this graph turns out to essentially be an “expander” in the sense that any two vertices can be connected (in multiple ways) by fairly short paths in this graph (if one allows one to modify one of or by ). As a consequence, we can assume that this quantity is essentially constant in (cf. the application of the ergodic theorem in this previous blog post), thus we now have

for most and some . By Taylor expansion, this implies that

on for most , thus

But this can be shown to contradict the Matomäki-Radziwill theorem (because the multiplicative function is known to be non-pretentious).

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