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I’ve just uploaded to the arXiv my paper “Almost all Collatz orbits attain almost bounded values“, submitted to the proceedings of the Forum of Mathematics, Pi. In this paper I returned to the topic of the notorious Collatz conjecture (also known as the conjecture), which I previously discussed in this blog post. This conjecture can be phrased as follows. Let denote the positive integers (with the natural numbers), and let be the map defined by setting equal to when is odd and when is even. Let be the minimal element of the Collatz orbit . Then we have

Conjecture 1 (Collatz conjecture)One has for all .

Establishing the conjecture for all remains out of reach of current techniques (for instance, as discussed in the previous blog post, it is basically at least as difficult as Baker’s theorem, all known proofs of which are quite difficult). However, the situation is more promising if one is willing to settle for results which only hold for “most” in some sense. For instance, it is a result of Krasikov and Lagarias that

for all sufficiently large . In another direction, it was shown by Terras that for almost all (in the sense of natural density), one has . This was then improved by Allouche to for almost all and any fixed , and extended later by Korec to cover all . In this paper we obtain the following further improvement (at the cost of weakening natural density to logarithmic density):

Theorem 2Let be any function with . Then we have for almost all (in the sense of logarithmic density).

Thus for instance one has for almost all (in the sense of logarithmic density).

The difficulty here is one usually only expects to establish “local-in-time” results that control the evolution for times that only get as large as a small multiple of ; the aforementioned results of Terras, Allouche, and Korec, for instance, are of this time. However, to get all the way down to one needs something more like an “(almost) global-in-time” result, where the evolution remains under control for so long that the orbit has nearly reached the bounded state .

However, as observed by Bourgain in the context of nonlinear Schrödinger equations, one can iterate “almost sure local wellposedness” type results (which give local control for almost all initial data from a given distribution) into “almost sure (almost) global wellposedness” type results if one is fortunate enough to draw one’s data from an *invariant measure* for the dynamics. To illustrate the idea, let us take Korec’s aforementioned result that if one picks at random an integer from a large interval , then in most cases, the orbit of will eventually move into the interval . Similarly, if one picks an integer at random from , then in most cases, the orbit of will eventually move into . It is then tempting to concatenate the two statements and conclude that for most in , the orbit will eventually move . Unfortunately, this argument does not quite work, because by the time the orbit from a randomly drawn reaches , the distribution of the final value is unlikely to be close to being uniformly distributed on , and in particular could potentially concentrate almost entirely in the exceptional set of that do not make it into . The point here is the uniform measure on is not transported by Collatz dynamics to anything resembling the uniform measure on .

So, one now needs to locate a measure which has better invariance properties under the Collatz dynamics. It turns out to be technically convenient to work with a standard acceleration of the Collatz map known as the *Syracuse map* , defined on the odd numbers by setting , where is the largest power of that divides . (The advantage of using the Syracuse map over the Collatz map is that it performs precisely one multiplication of at each iteration step, which makes the map better behaved when performing “-adic” analysis.)

When viewed -adically, we soon see that iterations of the Syracuse map become somewhat irregular. Most obviously, is never divisible by . A little less obviously, is twice as likely to equal mod as it is to equal mod . This is because for a randomly chosen odd , the number of times that divides can be seen to have a geometric distribution of mean – it equals any given value with probability . Such a geometric random variable is twice as likely to be odd as to be even, which is what gives the above irregularity. There are similar irregularities modulo higher powers of . For instance, one can compute that for large random odd , will take the residue classes with probabilities

respectively. More generally, for any , will be distributed according to the law of a random variable on that we call a *Syracuse random variable*, and can be described explicitly as

where are iid copies of a geometric random variable of mean .

In view of this, any proposed “invariant” (or approximately invariant) measure (or family of measures) for the Syracuse dynamics should take this -adic irregularity of distribution into account. It turns out that one can use the Syracuse random variables to construct such a measure, but only if these random variables stabilise in the limit in a certain total variation sense. More precisely, in the paper we establish the estimate

for any and any . This type of stabilisation is plausible from entropy heuristics – the tuple of geometric random variables that generates has Shannon entropy , which is significantly larger than the total entropy of the uniform distribution on , so we expect a lot of “mixing” and “collision” to occur when converting the tuple to ; these heuristics can be supported by numerics (which I was able to work out up to about before running into memory and CPU issues), but it turns out to be surprisingly delicate to make this precise.

A first hint of how to proceed comes from the elementary number theory observation (easily proven by induction) that the rational numbers

are all distinct as vary over tuples in . Unfortunately, the process of reducing mod creates a lot of collisions (as must happen from the pigeonhole principle); however, by a simple “Lefschetz principle” type argument one can at least show that the reductions

are mostly distinct for “typical” (as drawn using the geometric distribution) as long as is a bit smaller than (basically because the rational number appearing in (3) then typically takes a form like with an integer between and ). This analysis of the component (3) of (1) is already enough to get quite a bit of spreading on (roughly speaking, when the argument is optimised, it shows that this random variable cannot concentrate in any subset of of density less than for some large absolute constant ). To get from this to a stabilisation property (2) we have to exploit the mixing effects of the remaining portion of (1) that does not come from (3). After some standard Fourier-analytic manipulations, matters then boil down to obtaining non-trivial decay of the characteristic function of , and more precisely in showing that

for any and any that is not divisible by .

If the random variable (1) was the sum of independent terms, one could express this characteristic function as something like a Riesz product, which would be straightforward to estimate well. Unfortunately, the terms in (1) are loosely coupled together, and so the characteristic factor does not immediately factor into a Riesz product. However, if one groups adjacent terms in (1) together, one can rewrite it (assuming is even for sake of discussion) as

where . The point here is that after conditioning on the to be fixed, the random variables remain independent (though the distribution of each depends on the value that we conditioned to), and so the above expression is a *conditional* sum of independent random variables. This lets one express the characeteristic function of (1) as an *averaged* Riesz product. One can use this to establish the bound (4) as long as one can show that the expression

is not close to an integer for a moderately large number (, to be precise) of indices . (Actually, for technical reasons we have to also restrict to those for which , but let us ignore this detail here.) To put it another way, if we let denote the set of pairs for which

we have to show that (with overwhelming probability) the random walk

(which we view as a two-dimensional renewal process) contains at least a few points lying outside of .

A little bit of elementary number theory and combinatorics allows one to describe the set as the union of “triangles” with a certain non-zero separation between them. If the triangles were all fairly small, then one expects the renewal process to visit at least one point outside of after passing through any given such triangle, and it then becomes relatively easy to then show that the renewal process usually has the required number of points outside of . The most difficult case is when the renewal process passes through a particularly large triangle in . However, it turns out that large triangles enjoy particularly good separation properties, and in particular afer passing through a large triangle one is likely to only encounter nothing but small triangles for a while. After making these heuristics more precise, one is finally able to get enough points on the renewal process outside of that one can finish the proof of (4), and thus Theorem 2.

Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures“. This is a sequel to our previous paper that studied logarithmic correlations of the form

where were bounded multiplicative functions, were fixed shifts, was a quantity going off to infinity, and was a generalised limit functional. Our main technical result asserted that these correlations were necessarily the uniform limit of periodic functions . Furthermore, if (weakly) pretended to be a Dirichlet character , then the could be chosen to be –isotypic in the sense that whenever are integers with coprime to the periods of and ; otherwise, if did not weakly pretend to be any Dirichlet character , then vanished completely. This was then used to verify several cases of the logarithmically averaged Elliott and Chowla conjectures.

The purpose of this paper was to investigate the extent to which the methods could be extended to non-logarithmically averaged settings. For our main technical result, we now considered the unweighted averages

where is an additional parameter. Our main result was now as follows. If did not weakly pretend to be a twisted Dirichlet character , then converged to zero on (doubly logarithmic) average as . If instead did pretend to be such a twisted Dirichlet character, then converged on (doubly logarithmic) average to a limit of -isotypic functions . Thus, roughly speaking, one has the approximation

for most .

Informally, this says that at almost all scales (where “almost all” means “outside of a set of logarithmic density zero”), the non-logarithmic averages behave much like their logarithmic counterparts except for a possible additional twisting by an Archimedean character (which interacts with the Archimedean parameter in much the same way that the Dirichlet character interacts with the non-Archimedean parameter ). One consequence of this is that most of the recent results on the logarithmically averaged Chowla and Elliott conjectures can now be extended to their non-logarithmically averaged counterparts, so long as one excludes a set of exceptional scales of logarithmic density zero. For instance, the Chowla conjecture

is now established for either odd or equal to , so long as one excludes an exceptional set of scales.

In the logarithmically averaged setup, the main idea was to combine two very different pieces of information on . The first, coming from recent results in ergodic theory, was to show that was well approximated in some sense by a nilsequence. The second was to use the “entropy decrement argument” to obtain an approximate isotopy property of the form

for “most” primes and integers . Combining the two facts, one eventually finds that only the almost periodic components of the nilsequence are relevant.

In the current situation, each is approximated by a nilsequence, but the nilsequence can vary with (although there is some useful “Lipschitz continuity” of this nilsequence with respect to the parameter). Meanwhile, the entropy decrement argument gives an approximation basically of the form

for “most” . The arguments then proceed largely as in the logarithmically averaged case. A key lemma to handle the dependence on the new parameter is the following cohomological statement: if one has a map that was a quasimorphism in the sense that for all and some small , then there exists a real number such that for all small . This is achieved by applying a standard “cocycle averaging argument” to the cocycle .

It would of course be desirable to not have the set of exceptional scales. We only know of one (implausible) scenario in which we can do this, namely when one has far fewer (in particular, subexponentially many) sign patterns for (say) the Liouville function than predicted by the Chowla conjecture. In this scenario (roughly analogous to the “Siegel zero” scenario in multiplicative number theory), the entropy of the Liouville sign patterns is so small that the entropy decrement argument becomes powerful enough to control all scales rather than almost all scales. On the other hand, this scenario seems to be self-defeating, in that it allows one to establish a large number of cases of the Chowla conjecture, and the full Chowla conjecture is inconsistent with having unusually few sign patterns. Still it hints that future work in this direction may need to split into “low entropy” and “high entropy” cases, in analogy to how many arguments in multiplicative number theory have to split into the “Siegel zero” and “no Siegel zero” cases.

This coming fall quarter, I am teaching a class on topics in the mathematical theory of incompressible fluid equations, focusing particularly on the incompressible Euler and Navier-Stokes equations. These two equations are by no means the only equations used to model fluids, but I will focus on these two equations in this course to narrow the focus down to something manageable. I have not fully decided on the choice of topics to cover in this course, but I would probably begin with some core topics such as local well-posedness theory and blowup criteria, conservation laws, and construction of weak solutions, then move on to some topics such as boundary layers and the Prandtl equations, the Euler-Poincare-Arnold interpretation of the Euler equations as an infinite dimensional geodesic flow, and some discussion of the Onsager conjecture. I will probably also continue to more advanced and recent topics in the winter quarter.

In this initial set of notes, we begin by reviewing the physical derivation of the Euler and Navier-Stokes equations from the first principles of Newtonian mechanics, and specifically from Newton’s famous three laws of motion. Strictly speaking, this derivation is not needed for the mathematical analysis of these equations, which can be viewed if one wishes as an arbitrarily chosen system of partial differential equations without any physical motivation; however, I feel that the derivation sheds some insight and intuition on these equations, and is also worth knowing on purely intellectual grounds regardless of its mathematical consequences. I also find it instructive to actually see the journey from Newton’s law

to the seemingly rather different-looking law

for incompressible Navier-Stokes (or, if one drops the viscosity term , the Euler equations).

Our discussion in this set of notes is physical rather than mathematical, and so we will not be working at mathematical levels of rigour and precision. In particular we will be fairly casual about interchanging summations, limits, and integrals, we will manipulate approximate identities as if they were exact identities (e.g., by differentiating both sides of the approximate identity), and we will not attempt to verify any regularity or convergence hypotheses in the expressions being manipulated. (The same holds for the exercises in this text, which also do not need to be justified at mathematical levels of rigour.) Of course, once we resume the mathematical portion of this course in subsequent notes, such issues will be an important focus of careful attention. This is a basic division of labour in mathematical modeling: non-rigorous heuristic reasoning is used to derive a mathematical model from physical (or other “real-life”) principles, but once a precise model is obtained, the analysis of that model should be completely rigorous if at all possible (even if this requires applying the model to regimes which do not correspond to the original physical motivation of that model). See the discussion by John Ball quoted at the end of these slides of Gero Friesecke for an expansion of these points.

Note: our treatment here will differ slightly from that presented in many fluid mechanics texts, in that it will emphasise first-principles derivations from many-particle systems, rather than relying on bulk laws of physics, such as the laws of thermodynamics, which we will not cover here. (However, the derivations from bulk laws tend to be more robust, in that they are not as reliant on assumptions about the particular interactions between particles. In particular, the physical hypotheses we assume in this post are probably quite a bit stronger than the minimal assumptions needed to justify the Euler or Navier-Stokes equations, which can hold even in situations in which one or more of the hypotheses assumed here break down.)

Let be a measure-preserving system – a probability space equipped with a measure-preserving translation (which for simplicity of discussion we shall assume to be invertible). We will informally think of two points in this space as being “close” if for some that is not too large; this allows one to distinguish between “local” structure at a point (in which one only looks at nearby points for moderately large ) and “global” structure (in which one looks at the entire space ). The local/global distinction is also known as the time-averaged/space-averaged distinction in ergodic theory.

A measure-preserving system is said to be ergodic if all the invariant sets are either zero measure or full measure. An equivalent form of this statement is that any measurable function which is *locally essentially constant* in the sense that for -almost every , is necessarily *globally essentially constant* in the sense that there is a constant such that for -almost every . A basic consequence of ergodicity is the mean ergodic theorem: if , then the averages converge in norm to the mean . (The mean ergodic theorem also applies to other spaces with , though it is usually proven first in the Hilbert space .) Informally: in ergodic systems, time averages are asymptotically equal to space averages. Specialising to the case of indicator functions, this implies in particular that converges to for any measurable set .

In this short note I would like to use the mean ergodic theorem to show that ergodic systems also have the property that “somewhat locally constant” functions are necessarily “somewhat globally constant”; this is not a deep observation, and probably already in the literature, but I found it a cute statement that I had not previously seen. More precisely:

Corollary 1Let be an ergodic measure-preserving system, and let be measurable. Suppose that

for some . Then there exists a constant such that for in a set of measure at least .

Informally: if is locally constant on pairs at least of the time, then is globally constant at least of the time. Of course the claim fails if the ergodicity hypothesis is dropped, as one can simply take to be an invariant function that is not essentially constant, such as the indicator function of an invariant set of intermediate measure. This corollary can be viewed as a manifestation of the general principle that ergodic systems have the same “global” (or “space-averaged”) behaviour as “local” (or “time-averaged”) behaviour, in contrast to non-ergodic systems in which local properties do not automatically transfer over to their global counterparts.

*Proof:* By composing with (say) the arctangent function, we may assume without loss of generality that is bounded. Let , and partition as , where is the level set

For each , only finitely many of the are non-empty. By (1), one has

Using the ergodic theorem, we conclude that

On the other hand, . Thus there exists such that , thus

By the Bolzano-Weierstrass theorem, we may pass to a subsequence where converges to a limit , then we have

for infinitely many , and hence

The claim follows.

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 1Assume 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 sayfor 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 2Let 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 thatfor 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.

Let be a monic polynomial of degree with complex coefficients. Then by the fundamental theorem of algebra, we can factor as

for some complex zeroes (possibly with repetition).

Now suppose we evolve with respect to time by heat flow, creating a function of two variables with given initial data for which

On the space of polynomials of degree at most , the operator is nilpotent, and one can solve this equation explicitly both forwards and backwards in time by the Taylor series

For instance, if one starts with a quadratic , then the polynomial evolves by the formula

As the polynomial evolves in time, the zeroes evolve also. Assuming for sake of discussion that the zeroes are simple, the inverse function theorem tells us that the zeroes will (locally, at least) evolve smoothly in time. What are the dynamics of this evolution?

For instance, in the quadratic case, the quadratic formula tells us that the zeroes are

and

after arbitrarily choosing a branch of the square root. If are real and the discriminant is initially positive, we see that we start with two real zeroes centred around , which then approach each other until time , at which point the roots collide and then move off from each other in an imaginary direction.

In the general case, we can obtain the equations of motion by implicitly differentiating the defining equation

in time using (2) to obtain

To simplify notation we drop the explicit dependence on time, thus

From (1) and the product rule, we see that

and

(where all indices are understood to range over ) leading to the equations of motion

at least when one avoids those times in which there is a repeated zero. In the case when the zeroes are real, each term represents a (first-order) attraction in the dynamics between and , but the dynamics are more complicated for complex zeroes (e.g. purely imaginary zeroes will experience repulsion rather than attraction, as one already sees in the quadratic example). Curiously, this system resembles that of Dyson brownian motion (except with the brownian motion part removed, and time reversed). I learned of the connection between the ODE (3) and the heat equation from this paper of Csordas, Smith, and Varga, but perhaps it has been mentioned in earlier literature as well.

One interesting consequence of these equations is that if the zeroes are real at some time, then they will stay real as long as the zeroes do not collide. Let us now restrict attention to the case of real simple zeroes, in which case we will rename the zeroes as instead of , and order them as . The evolution

can now be thought of as reverse gradient flow for the “entropy”

(which is also essentially the logarithm of the discriminant of the polynomial) since we have

In particular, we have the monotonicity formula

where is the “energy”

where in the last line we use the antisymmetrisation identity

Among other things, this shows that as one goes backwards in time, the entropy decreases, and so no collisions can occur to the past, only in the future, which is of course consistent with the attractive nature of the dynamics. As is a convex function of the positions , one expects to also evolve in a convex manner in time, that is to say the energy should be increasing. This is indeed the case:

Exercise 1Show that

Symmetric polynomials of the zeroes are polynomial functions of the coefficients and should thus evolve in a polynomial fashion. One can compute this explicitly in simple cases. For instance, the center of mass is an invariant:

The variance decreases linearly:

Exercise 2Establish the virial identity

As the variance (which is proportional to ) cannot become negative, this identity shows that “finite time blowup” must occur – that the zeroes must collide at or before the time .

Exercise 3Show that theStieltjes transformsolves the viscous Burgers equation

either by using the original heat equation (2) and the identity , or else by using the equations of motion (3). This relation between the Burgers equation and the heat equation is known as the Cole-Hopf transformation.

The paper of Csordas, Smith, and Varga mentioned previously gives some other bounds on the lifespan of the dynamics; roughly speaking, they show that if there is one pair of zeroes that are much closer to each other than to the other zeroes then they must collide in a short amount of time (unless there is a collision occuring even earlier at some other location). Their argument extends also to situations where there are an infinite number of zeroes, which they apply to get new results on Newman’s conjecture in analytic number theory. I would be curious to know of further places in the literature where this dynamics has been studied.

Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures“, submitted to Duke Mathematical Journal. This paper builds upon my previous paper in which I introduced an “entropy decrement method” to prove the two-point (logarithmically averaged) cases of the Chowla and Elliott conjectures. A bit more specifically, I showed that

whenever were sequences going to infinity, were distinct integers, and were -bounded multiplicative functions which were *non-pretentious* in the sense that

for all Dirichlet characters and for . Thus, for instance, one had the logarithmically averaged two-point Chowla conjecture

for fixed any non-zero , where was the Liouville function.

One would certainly like to extend these results to higher order correlations than the two-point correlations. This looks to be difficult (though perhaps not completely impossible if one allows for logarithmic averaging): in a previous paper I showed that achieving this in the context of the Liouville function would be equivalent to resolving the logarithmically averaged Sarnak conjecture, as well as establishing logarithmically averaged local Gowers uniformity of the Liouville function. However, in this paper we are able to avoid having to resolve these difficult conjectures to obtain partial results towards the (logarithmically averaged) Chowla and Elliott conjecture. For the Chowla conjecture, we can obtain all odd order correlations, in that

for all odd and all integers (which, in the odd order case, are no longer required to be distinct). (Superficially, this looks like we have resolved “half of the cases” of the logarithmically averaged Chowla conjecture; but it seems the odd order correlations are significantly easier than the even order ones. For instance, because of the Katai-Bourgain-Sarnak-Ziegler criterion, one can basically deduce the odd order cases of (2) from the even order cases (after allowing for some dilations in the argument ).

For the more general Elliott conjecture, we can show that

for any , any integers and any bounded multiplicative functions , unless the product *weakly pretends to be a Dirichlet character * in the sense that

This can be seen to imply (2) as a special case. Even when *does* pretend to be a Dirichlet character , we can still say something: if the limits

exist for each (which can be guaranteed if we pass to a suitable subsequence), then is the uniform limit of periodic functions , each of which is –isotypic in the sense that whenever are integers with coprime to the periods of and . This does not pin down the value of any single correlation , but does put significant constraints on how these correlations may vary with .

Among other things, this allows us to show that all possible length four sign patterns of the Liouville function occur with positive density, and all possible length four sign patterns occur with the conjectured logarithmic density. (In a previous paper with Matomaki and Radziwill, we obtained comparable results for length three patterns of Liouville and length two patterns of Möbius.)

To describe the argument, let us focus for simplicity on the case of the Liouville correlations

assuming for sake of discussion that all limits exist. (In the paper, we instead use the device of generalised limits, as discussed in this previous post.) The idea is to combine together two rather different ways to control this function . The first proceeds by the entropy decrement method mentioned earlier, which roughly speaking works as follows. Firstly, we pick a prime and observe that for any , which allows us to rewrite (3) as

Making the change of variables , we obtain

The difference between and is negligible in the limit (here is where we crucially rely on the log-averaging), hence

and thus by (3) we have

The entropy decrement argument can be used to show that the latter limit is small for most (roughly speaking, this is because the factors behave like independent random variables as varies, so that concentration of measure results such as Hoeffding’s inequality can apply, after using entropy inequalities to decouple somewhat these random variables from the factors). We thus obtain the approximate isotopy property

On the other hand, by the Furstenberg correspondence principle (as discussed in these previous posts), it is possible to express as a multiple correlation

for some probability space equipped with a measure-preserving invertible map . Using results of Bergelson-Host-Kra, Leibman, and Le, this allows us to obtain a decomposition of the form

where is a nilsequence, and goes to zero in density (even along the primes, or constant multiples of the primes). The original work of Bergelson-Host-Kra required ergodicity on , which is very definitely a hypothesis that is not available here; however, the later work of Leibman removed this hypothesis, and the work of Le refined the control on so that one still has good control when restricting to primes, or constant multiples of primes.

Ignoring the small error , we can now combine (5) to conclude that

Using the equidistribution theory of nilsequences (as developed in this previous paper of Ben Green and myself), one can break up further into a periodic piece and an “irrational” or “minor arc” piece . The contribution of the minor arc piece can be shown to mostly cancel itself out after dilating by primes and averaging, thanks to Vinogradov-type bilinear sum estimates (transferred to the primes). So we end up with

which already shows (heuristically, at least) the claim that can be approximated by periodic functions which are isotopic in the sense that

But if is odd, one can use Dirichlet’s theorem on primes in arithmetic progressions to restrict to primes that are modulo the period of , and conclude now that vanishes identically, which (heuristically, at least) gives (2).

The same sort of argument works to give the more general bounds on correlations of bounded multiplicative functions. But for the specific task of proving (2), we initially used a slightly different argument that avoids using the ergodic theory machinery of Bergelson-Host-Kra, Leibman, and Le, but replaces it instead with the Gowers uniformity norm theory used to count linear equations in primes. Basically, by averaging (4) in using the “-trick”, as well as known facts about the Gowers uniformity of the von Mangoldt function, one can obtain an approximation of the form

where ranges over a large range of integers coprime to some primorial . On the other hand, by iterating (4) we have

for most semiprimes , and by again averaging over semiprimes one can obtain an approximation of the form

For odd, one can combine the two approximations to conclude that . (This argument is not given in the current paper, but we plan to detail it in a subsequent one.)

I’ve just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds“, submitted to Discrete and Continuous Dynamical Systems. This is a variant of my recent paper on the universality of potential well dynamics, but instead of trying to embed dynamical systems into a potential well , here we try to embed dynamical systems into the incompressible Euler equations

on a Riemannian manifold . (One is particularly interested in the case of flat manifolds , particularly or , but for the main result of this paper it is essential that one is permitted to consider curved manifolds.) This system, first studied by Ebin and Marsden, is the natural generalisation of the usual incompressible Euler equations to curved space; it can be viewed as the formal geodesic flow equation on the infinite-dimensional manifold of volume-preserving diffeomorphisms on (see this previous post for a discussion of this in the flat space case).

The Euler equations can be viewed as a nonlinear equation in which the nonlinearity is a quadratic function of the velocity field . It is thus natural to compare the Euler equations with quadratic ODE of the form

where is the unknown solution, and is a bilinear map, which we may assume without loss of generality to be symmetric. One can ask whether such an ODE may be linearly embedded into the Euler equations on some Riemannian manifold , which means that there is an injective linear map from to smooth vector fields on , as well as a bilinear map to smooth scalar fields on , such that the map takes solutions to (2) to solutions to (1), or equivalently that

for all .

For simplicity let us restrict to be compact. There is an obvious necessary condition for this embeddability to occur, which comes from energy conservation law for the Euler equations; unpacking everything, this implies that the bilinear form in (2) has to obey a cancellation condition

for some positive definite inner product on . The main result of the paper is the converse to this statement: if is a symmetric bilinear form obeying a cancellation condition (3), then it is possible to embed the equations (2) into the Euler equations (1) on some Riemannian manifold ; the catch is that this manifold will depend on the form and on the dimension (in fact in the construction I have, is given explicitly as , with a funny metric on it that depends on ).

As a consequence, any finite dimensional portion of the usual “dyadic shell models” used as simplified toy models of the Euler equation, can actually be embedded into a genuine Euler equation, albeit on a high-dimensional and curved manifold. This includes portions of the self-similar “machine” I used in a previous paper to establish finite time blowup for an averaged version of the Navier-Stokes (or Euler) equations. Unfortunately, the result in this paper does not apply to infinite-dimensional ODE, so I cannot yet establish finite time blowup for the Euler equations on a (well-chosen) manifold. It does not seem so far beyond the realm of possibility, though, that this could be done in the relatively near future. In particular, the result here suggests that one could construct something resembling a universal Turing machine within an Euler flow on a manifold, which was one ingredient I would need to engineer such a finite time blowup.

The proof of the main theorem proceeds by an “elimination of variables” strategy that was used in some of my previous papers in this area, though in this particular case the Nash embedding theorem (or variants thereof) are not required. The first step is to lessen the dependence on the metric by partially reformulating the Euler equations (1) in terms of the covelocity (which is a -form) instead of the velocity . Using the freedom to modify the dimension of the underlying manifold , one can also decouple the metric from the volume form that is used to obtain the divergence-free condition. At this point the metric can be eliminated, with a certain positive definiteness condition between the velocity and covelocity taking its place. After a substantial amount of trial and error (motivated by some “two-and-a-half-dimensional” reductions of the three-dimensional Euler equations, and also by playing around with a number of variants of the classic “separation of variables” strategy), I eventually found an ansatz for the velocity and covelocity that automatically solved most of the components of the Euler equations (as well as most of the positive definiteness requirements), as long as one could find a number of scalar fields that obeyed a certain nonlinear system of transport equations, and also obeyed a positive definiteness condition. Here I was stuck for a bit because the system I ended up with was overdetermined – more equations than unknowns. After trying a number of special cases I eventually found a solution to the transport system on the sphere, except that the scalar functions sometimes degenerated and so the positive definiteness property I wanted was only obeyed with positive semi-definiteness. I tried for some time to perturb this example into a strictly positive definite solution before eventually working out that this was not possible. Finally I had the brainwave to lift the solution from the sphere to an even more symmetric space, and this quickly led to the final solution of the problem, using the special orthogonal group rather than the sphere as the underlying domain. The solution ended up being rather simple in form, but it is still somewhat miraculous to me that it exists at all; in retrospect, given the overdetermined nature of the problem, relying on a large amount of symmetry to cut down the number of equations was basically the only hope.

I’ve just uploaded to the arXiv my paper “On the universality of potential well dynamics“, submitted to Dynamics of PDE. This is a spinoff from my previous paper on blowup of nonlinear wave equations, inspired by some conversations with Sungjin Oh. Here we focus mainly on the zero-dimensional case of such equations, namely the potential well equation

for a particle trapped in a potential well with potential , with as . This ODE always admits global solutions from arbitrary initial positions and initial velocities , thanks to conservation of the Hamiltonian . As this Hamiltonian is coercive (in that its level sets are compact), solutions to this equation are always almost periodic. On the other hand, as can already be seen using the harmonic oscillator (and direct sums of this system), this equation can generate periodic solutions, as well as quasiperiodic solutions.

All quasiperiodic motions are almost periodic. However, there are many examples of dynamical systems that admit solutions that are almost periodic but not quasiperiodic. So one can pose the question: are the dynamics of potential wells *universal* in the sense that they can capture all almost periodic solutions?

A precise question can be phrased as follows. Let be a compact manifold, and let be a smooth vector field on ; to avoid degeneracies, let us take to be *non-singular* in the sense that it is everywhere non-vanishing. Then the trajectories of the first-order ODE

for are always global and almost periodic. Can we then find a (coercive) potential for some , as well as a smooth embedding , such that every solution to (2) pushes forward under to a solution to (1)? (Actually, for technical reasons it is preferable to map into the phase space , rather than position space , but let us ignore this detail for this discussion.)

It turns out that the answer is no; there is a very specific obstruction. Given a pair as above, define a *strongly adapted -form* to be a -form on such that is pointwise positive, and the Lie derivative is an exact -form. We then have

Theorem 1A smooth compact non-singular dynamics can be embedded smoothly in a potential well system if and only if it admits a strongly adapted -form.

For the “only if” direction, the key point is that potential wells (viewed as a Hamiltonian flow on the phase space ) admit a strongly adapted -form, namely the canonical -form , whose Lie derivative is the derivative of the Lagrangian and is thus exact. The converse “if” direction is mainly a consequence of the Nash embedding theorem, and follows the arguments used in my previous paper.

Interestingly, the same obstruction also works for potential wells in a more general Riemannian manifold than , or for nonlinear wave equations with a potential; combining the two, the obstruction is also present for wave maps with a potential.

It is then natural to ask whether this obstruction is non-trivial, in the sense that there are at least some examples of dynamics that do not support strongly adapted -forms (and hence cannot be modeled smoothly by the dynamics of a potential well, nonlinear wave equation, or wave maps). I posed this question on MathOverflow, and Robert Bryant provided a very nice construction, showing that the vector field on the -torus had no strongly adapted -forms, and hence the dynamics of this vector field cannot be smoothly reproduced by a potential well, nonlinear wave equation, or wave map:

On the other hand, the suspension of any diffeomorphism does support a strongly adapted -form (the derivative of the time coordinate), and using this and the previous theorem I was able to embed a universal Turing machine into a potential well. In particular, there are flows for an explicitly describable potential well whose trajectories have behavior that is undecidable using the usual ZFC axioms of set theory! So potential well dynamics are “effectively” universal, despite the presence of the aforementioned obstruction.

In my previous work on blowup for Navier-Stokes like equations, I speculated that if one could somehow replicate a universal Turing machine within the Euler equations, one could use this machine to create a “von Neumann machine” that replicated smaller versions of itself, which on iteration would lead to a finite time blowup. Now that such a mechanism is present in nonlinear wave equations, it is tempting to try to make this scheme work in that setting. Of course, in my previous paper I had already demonstrated finite time blowup, at least in a three-dimensional setting, but that was a relatively simple discretely self-similar blowup in which no computation occurred. This more complicated blowup scheme would be significantly more effort to set up, but would be proof-of-concept that the same scheme would in principle be possible for the Navier-Stokes equations, assuming somehow that one can embed a universal Turing machine into the Euler equations. (But I’m still hopelessly stuck on how to accomplish this latter task…)

A sequence of complex numbers is said to be quasiperiodic if it is of the form

for some real numbers and continuous function . For instance, linear phases such as (where ) are examples of quasiperiodic sequences; the top order coefficient (modulo ) can be viewed as a “frequency” of the integers, and an element of the Pontryagin dual of the integers. Any periodic sequence is also quasiperiodic (taking and to be the reciprocal of the period). A sequence is said to be almost periodic if it is the uniform limit of quasiperiodic sequences. For instance any Fourier series of the form

with real numbers and an absolutely summable sequence of complex coefficients, will be almost periodic.

These sequences arise in various “complexity one” problems in arithmetic combinatorics and ergodic theory. For instance, if is a measure-preserving system – a probability space equipped with a measure-preserving shift, and are bounded measurable functions, then the correlation sequence

can be shown to be an almost periodic sequence, plus an error term which is “null” in the sense that it has vanishing uniform density:

This can be established in a number of ways, for instance by writing as the Fourier coefficients of the spectral measure of the shift with respect to the functions , and then decomposing that measure into pure point and continuous components.

In the last two decades or so, it has become clear that there are natural higher order versions of these concepts, in which linear polynomials such as are replaced with higher degree counterparts. The most obvious candidates for these counterparts would be the polynomials , but this turns out to not be a complete set of higher degree objects needed for the theory. Instead, the higher order versions of quasiperiodic and almost periodic sequences are now known as *basic nilsequences* and *nilsequences* respectively, while the higher order version of a linear phase is a *nilcharacter*; each nilcharacter then has a *symbol* that is a higher order generalisation of the concept of a frequency (and the collection of all symbols forms a group that can be viewed as a higher order version of the Pontryagin dual of ). The theory of these objects is spread out in the literature across a number of papers; in particular, the theory of nilcharacters is mostly developed in Appendix E of this 116-page paper of Ben Green, Tamar Ziegler, and myself, and is furthermore written using nonstandard analysis and treating the more general setting of higher dimensional sequences. I therefore decided to rewrite some of that material in this blog post, in the simpler context of the qualitative asymptotic theory of one-dimensional nilsequences and nilcharacters rather than the quantitative single-scale theory that is needed for combinatorial applications (and which necessitated the use of nonstandard analysis in the previous paper).

For technical reasons (having to do with the non-trivial topological structure on nilmanifolds), it will be convenient to work with vector-valued sequences, that take values in a finite-dimensional complex vector space rather than in . By doing so, the space of sequences is now, technically, no longer a ring, as the operations of addition and multiplication on vector-valued sequences become ill-defined. However, we can still take complex conjugates of a sequence, and add sequences taking values in the same vector space , and for sequences taking values in different vector spaces , , we may utilise the tensor product , which we will normalise by defining

This product is associative and bilinear, and also commutative up to permutation of the indices. It also interacts well with the Hermitian norm

since we have .

The traditional definition of a basic nilsequence (as defined for instance by Bergelson, Host, and Kra) is as follows:

Definition 1 (Basic nilsequence, first definition)Anilmanifold of step at mostis a quotient , where is a connected, simply connected nilpotent Lie group of step at most (thus, all -fold commutators vanish) and is a discrete cocompact lattice in . Abasic nilsequence of degree at mostis a sequence of the form , where , , and is a continuous function.

For instance, it is not difficult using this definition to show that a sequence is a basic nilsequence of degree at most if and only if it is quasiperiodic. The requirement that be simply connected can be easily removed if desired by passing to a universal cover, but it is technically convenient to assume it (among other things, it allows for a well-defined logarithm map that obeys the Baker-Campbell-Hausdorff formula). When one wishes to perform a more quantitative analysis of nilsequences (particularly when working on a “single scale”. sich as on a single long interval ), it is common to impose additional regularity conditions on the function , such as Lipschitz continuity or smoothness, but ordinary continuity will suffice for the qualitative discussion in this blog post.

Nowadays, and particularly when one needs to understand the “single-scale” equidistribution properties of nilsequences, it is more convenient (as is for instance done in this ICM paper of Green) to use an alternate definition of a nilsequence as follows.

Definition 2Let . Afiltered group of degree at mostis a group together with a sequence of subgroups with and for . Apolynomial sequenceinto a filtered group is a function such that for all and , where is the difference operator. Afiltered nilmanifold of degree at mostis a quotient , where is a filtered group of degree at most such that and all of the subgroups are connected, simply connected nilpotent filtered Lie group, and is a discrete cocompact subgroup of such that is a discrete cocompact subgroup of . Abasic nilsequence of degree at mostis a sequence of the form , where is a polynomial sequence, is a filtered nilmanifold of degree at most , and is a continuous function which is -automorphic, in the sense that for all and .

One can easily identify a -automorphic function on with a function on , but there are some (very minor) advantages to working on the group instead of the quotient , as it becomes slightly easier to modify the automorphy group when needed. (But because the action of on is free, one can pass from -automorphic functions on to functions on with very little difficulty.) The main reason to work with polynomial sequences rather than geometric progressions is that they form a group, a fact essentially established by by Lazard and Leibman; see Corollary B.4 of this paper of Green, Ziegler, and myself for a proof in the filtered group setting.

It is easy to see that any sequence that is a basic nilsequence of degree at most in the sense of the first definition, is also a basic nilsequence of degree at most in the second definition, since a nilmanifold of degree at most can be filtered using the lower central series, and any linear sequence will be a polynomial sequence with respect to that filtration. The converse implication is a little trickier, but still not too hard to show: see Appendix C of this paper of Ben Green, Tamar Ziegler, and myself. There are two key examples of basic nilsequences to keep in mind. The first are the polynomially quasiperiodic sequences

where are polynomials of degree at most , and is a -automorphic (i.e., -periodic) continuous function. The map defined by is a polynomial map of degree at most , if one filters by defining to equal when , and for . The torus then becomes a filtered nilmanifold of degree at most , and is thus a basic nilsequence of degree at most as per the second definition. It is also possible explicitly describe as a basic nilsequence of degree at most as per the first definition, for instance (in the case) by taking to be the space of upper triangular unipotent real matrices, and the subgroup with integer coefficients; we leave the details to the interested reader.

The other key example is a sequence of the form

where are real numbers, denotes the fractional part of , and and is a -automorphic continuous function that vanishes in a neighbourhood of . To describe this as a nilsequence, we use the nilpotent connected, simply connected degree , Heisenberg group

with the lower central series filtration , , and for , to be the discrete compact subgroup

to be the polynomial sequence

and to be the -automorphic function

one easily verifies that this function is indeed -automorphic, and it is continuous thanks to the vanishing properties of . Also we have , so is a basic nilsequence of degree at most . One can concoct similar examples with replaced by other “bracket polynomials” of ; for instance

will be a basic nilsequence if now vanishes in a neighbourhood of rather than . See this paper of Bergelson and Leibman for more discussion of bracket polynomials (also known as generalised polynomials) and their relationship to nilsequences.

A *nilsequence of degree at most * is defined to be a sequence that is the uniform limit of basic nilsequences of degree at most . Thus for instance a sequence is a nilsequence of degree at most if and only if it is almost periodic, while a sequence is a nilsequence of degree at most if and only if it is constant. Such objects arise in higher order recurrence: for instance, if are integers, is a measure-preserving system, and , then it was shown by Leibman that the sequence

is equal to a nilsequence of degree at most , plus a null sequence. (The special case when the measure-preserving system was ergodic and for was previously established by Bergelson, Host, and Kra.) Nilsequences also arise in the inverse theory of the Gowers uniformity norms, as discussed for instance in this previous post.

It is easy to see that a sequence is a basic nilsequence of degree at most if and only if each of its components are. The scalar basic nilsequences of degree are easily seen to form a -algebra (that is to say, they are a complex vector space closed under pointwise multiplication and complex conjugation), which implies similarly that vector-valued basic nilsequences of degree at most form a complex vector space closed under complex conjugation for each , and that the tensor product of any two basic nilsequences of degree at most is another basic nilsequence of degree at most . Similarly with “basic nilsequence” replaced by “nilsequence” throughout.

Now we turn to the notion of a nilcharacter, as defined in this paper of Ben Green, Tamar Ziegler, and myself:

Definition 3 (Nilcharacters)Let . Asub-nilcharacter of degreeis a basic nilsequence of degree at most , such that obeys the additional modulation propertyfor all and , where is a continuous homomorphism . (Note from (1) and -automorphy that unless vanishes identically, must map to , thus without loss of generality one can view as an element of the Pontryagial dual of the torus .) If in addition one has for all , we call a

nilcharacterof degree .

In the degree one case , the only sub-nilcharacters are of the form for some vector and , and this is a nilcharacter if is a unit vector. Similarly, in higher degree, any sequence of the form , where is a vector and is a polynomial of degree at most , is a sub-nilcharacter of degree , and a character if is a unit vector. A nilsequence of degree at most is automatically a sub-nilcharacter of degree , and a nilcharacter if it is of magnitude . A further example of a nilcharacter is provided by the two-dimensional sequence defined by

where are continuous, -automorphic functions that vanish on a neighbourhood of and respectively, and which form a partition of unity in the sense that

for all . Note that one needs both and to be not identically zero in order for all these conditions to be satisfied; it turns out (for topological reasons) that there is no scalar nilcharacter that is “equivalent” to this nilcharacter in a sense to be defined shortly. In some literature, one works exclusively with sub-nilcharacters rather than nilcharacters, however the former space contains zero-divisors, which is a little annoying technically. Nevertheless, both nilcharacters and sub-nilcharacters generate the same set of “symbols” as we shall see later.

We claim that every degree sub-nilcharacter can be expressed in the form , where is a degree nilcharacter, and is a linear transformation. Indeed, by scaling we may assume where uniformly. Using partitions of unity, one can find further functions also obeying (1) for the same character such that is non-zero; by dividing out the by the square root of this quantity, and then multiplying by , we may assume that

and then

becomes a degree nilcharacter that contains amongst its components, giving the claim.

As we shall show below, nilsequences can be approximated uniformly by linear combinations of nilcharacters, in much the same way that quasiperiodic or almost periodic sequences can be approximated uniformly by linear combinations of linear phases. In particular, nilcharacters can be used as “obstructions to uniformity” in the sense of the inverse theory of the Gowers uniformity norms.

The space of degree nilcharacters forms a semigroup under tensor product, with the constant sequence as the identity. One can upgrade this semigroup to an abelian group by quotienting nilcharacters out by equivalence:

Definition 4Let . We say that two degree nilcharacters , areequivalentif is equal (as a sequence) to a basic nilsequence of degree at most . (We will later show that this is indeed an equivalence relation.) The equivalence class of such a nilcharacter will be called thesymbolof that nilcharacter (in analogy to the symbol of a differential or pseudodifferential operator), and the collection of such symbols will be denoted .

As we shall see below the fold, has the structure of an abelian group, and enjoys some nice “symbol calculus” properties; also, one can view symbols as precisely describing the obstruction to equidistribution for nilsequences. For , the group is isomorphic to the Ponytragin dual of the integers, and for should be viewed as higher order generalisations of this Pontryagin dual. In principle, this group can be explicitly described for all , but the theory rapidly gets complicated as increases (much as the classification of nilpotent Lie groups or Lie algebras of step rapidly gets complicated even for medium-sized such as or ). We will give an explicit description of the case here. There is however one nice (and non-trivial) feature of for – it is not just an abelian group, but is in fact a vector space over the rationals !

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