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Define the *Collatz map* on the natural numbers by setting to equal when is odd and when is even, and let denote the forward Collatz orbit of . The notorious Collatz conjecture asserts that for all . Equivalently, if we define the backwards Collatz orbit to be all the natural numbers that encounter in their forward Collatz orbit, then the Collatz conjecture asserts that . As a partial result towards this latter statement, Krasikov and Lagarias in 2003 established the bound

for all and . (This improved upon previous values of obtained by Applegate and Lagarias in 1995, by Applegate and Lagarias in 1995 by a different method, by Wirsching in 1993, by Krasikov in 1989, by Sander in 1990, and some by Crandall in 1978.) This is still the largest value of for which (1) has been established. Of course, the Collatz conjecture would imply that we can take equal to , which is the assertion that a positive density set of natural numbers obeys the Collatz conjecture. This is not yet established, although the results in my previous paper do at least imply that a positive density set of natural numbers iterates to an (explicitly computable) bounded set, so in principle the case of (1) could now be verified by an (enormous) finite computation in which one verifies that every number in this explicit bounded set iterates to . In this post I would like to record a possible alternate route to this problem that depends on the distribution of a certain family of random variables that appeared in my previous paper, that I called *Syracuse random variables*.

Definition 1 (Syracuse random variables)For any natural number , aSyracuse random variableon the cyclic group is defined as a random variable of the form

where are independent copies of a geometric random variable on the natural numbers with mean , thus

} for . In (2) the arithmetic is performed in the ring .

Thus for instance

and so forth. After reversing the labeling of the , one could also view as the mod reduction of a -adic random variable

The probability density function of the Syracuse random variable can be explicitly computed by a recursive formula (see Lemma 1.12 of my previous paper). For instance, when , is equal to for respectively, while when , is equal to

when respectively.

The relationship of these random variables to the Collatz problem can be explained as follows. Let denote the odd natural numbers, and define the *Syracuse map* by

where the –valuation is the number of times divides . We can define the forward orbit and backward orbit of the Syracuse map as before. It is not difficult to then see that the Collatz conjecture is equivalent to the assertion , and that the assertion (1) for a given is equivalent to the assertion

for all , where is now understood to range over odd natural numbers. A brief calculation then shows that for any odd natural number and natural number , one has

where the natural numbers are defined by the formula

so in particular

Heuristically, one expects the -valuation of a typical odd number to be approximately distributed according to the geometric distribution , so one therefore expects the residue class to be distributed approximately according to the random variable .

The Syracuse random variables will always avoid multiples of three (this reflects the fact that is never a multiple of three), but attains any non-multiple of three in with positive probability. For any natural number , set

Equivalently, is the greatest quantity for which we have the inequality

for all integers not divisible by three, where is the set of all tuples for which

Thus for instance , , and . On the other hand, since all the probabilities sum to as ranges over the non-multiples of , we have the trivial upper bound

There is also an easy submultiplicativity result:

*Proof:* Let be an integer not divisible by , then by (4) we have

If we let denote the set of tuples that can be formed from the tuples in by deleting the final component from each tuple, then we have

with an integer not divisible by three. By definition of and a relabeling, we then have

for all . For such tuples we then have

so that . Since

for each , the claim follows.

From this lemma we see that for some absolute constant . Heuristically, we expect the Syracuse random variables to be somewhat approximately equidistributed amongst the multiples of (in Proposition 1.4 of my previous paper I prove a fine scale mixing result that supports this heuristic). As a consequence it is natural to conjecture that . I cannot prove this, but I can show that this conjecture would imply that we can take the exponent in (1), (3) arbitrarily close to one:

Proposition 3Suppose that (that is to say, as ). Thenas , or equivalently

I prove this proposition below the fold. A variant of the argument shows that for any value of , (1), (3) holds whenever , where is an explicitly computable function with as . In principle, one could then improve the Krasikov-Lagarias result by getting a sufficiently good upper bound on , which is in principle achievable numerically (note for instance that Lemma 2 implies the bound for any , since for any ).

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 !

The equidistribution theorem asserts that if is an irrational phase, then the sequence is equidistributed on the unit circle, or equivalently that

for any continuous (or equivalently, for any smooth) function . By approximating uniformly by a Fourier series, this claim is equivalent to that of showing that

for any non-zero integer (where ), which is easily verified from the irrationality of and the geometric series formula. Conversely, if is rational, then clearly fails to go to zero when is a multiple of the denominator of .

One can then ask for more quantitative information about the decay of exponential sums of , or more generally on exponential sums of the form for an arithmetic progression (in this post all progressions are understood to be finite) and a polynomial . It will be convenient to phrase such information in the form of an *inverse theorem*, describing those phases for which the exponential sum is large. Indeed, we have

Lemma 1 (Geometric series formula, inverse form)Let be an arithmetic progression of length at most for some , and let be a linear polynomial for some . Iffor some , then there exists a subprogression of of size such that varies by at most on (that is to say, lies in a subinterval of of length at most ).

*Proof:* By a linear change of variable we may assume that is of the form for some . We may of course assume that is non-zero in , so that ( denotes the distance from to the nearest integer). From the geometric series formula we see that

and so . Setting for some sufficiently small absolute constant , we obtain the claim.

Thus, in order for a linear phase to fail to be equidistributed on some long progression , must in fact be almost constant on large piece of .

As is well known, this phenomenon generalises to higher order polynomials. To achieve this, we need two elementary additional lemmas. The first relates the exponential sums of to the exponential sums of its “first derivatives” .

Lemma 2 (Van der Corput lemma, inverse form)Let be an arithmetic progression of length at most , and let be an arbitrary function such that

for some . Then, for integers , there exists a subprogression of , of the same spacing as , such that

*Proof:* Squaring (1), we see that

We write and conclude that

where is a subprogression of of the same spacing. Since , we conclude that

for values of (this can be seen, much like the pigeonhole principle, by arguing via contradiction for a suitable choice of implied constants). The claim follows.

The second lemma (which we recycle from this previous blog post) is a variant of the equidistribution theorem.

Lemma 3 (Vinogradov lemma)Let be an interval for some , and let be such that for at least values of , for some . Then eitheror

or else there is a natural number such that

*Proof:* We may assume that and , since we are done otherwise. Then there are at least two with , and by the pigeonhole principle we can find in with and . By the triangle inequality, we conclude that there exists at least one natural number for which

We take to be minimal amongst all such natural numbers, then we see that there exists coprime to and such that

If then we are done, so suppose that . Suppose that are elements of such that and . Writing for some , we have

By hypothesis, ; note that as and we also have . This implies that and thus . We then have

We conclude that for fixed with , there are at most elements of such that . Iterating this with a greedy algorithm, we see that the number of with is at most ; since , this implies that

and the claim follows.

Now we can quickly obtain a higher degree version of Lemma 1:

Proposition 4 (Weyl exponential sum estimate, inverse form)Let be an arithmetic progression of length at most for some , and let be a polynomial of some degree at most . Iffor some , then there exists a subprogression of with such that varies by at most on .

*Proof:* We induct on . The cases are immediate from Lemma 1. Now suppose that , and that the claim had already been proven for . To simplify the notation we allow implied constants to depend on . Let the hypotheses be as in the proposition. Clearly cannot exceed . By shrinking as necessary we may assume that for some sufficiently small constant depending on .

By rescaling we may assume . By Lemma 3, we see that for choices of such that

for some interval . We write , then is a polynomial of degree at most with leading coefficient . We conclude from induction hypothesis that for each such , there exists a natural number such that , by double-counting, this implies that there are integers in the interval such that . Applying Lemma 3, we conclude that either , or that

In the former case the claim is trivial (just take to be a point), so we may assume that we are in the latter case.

We partition into arithmetic progressions of spacing and length comparable to for some large depending on to be chosen later. By hypothesis, we have

so by the pigeonhole principle, we have

for at least one such progression . On this progression, we may use the binomial theorem and (4) to write as a polynomial in of degree at most , plus an error of size . We thus can write for for some polynomial of degree at most . By the triangle inequality, we thus have (for large enough) that

and hence by induction hypothesis we may find a subprogression of of size such that varies by most on , and thus (for large enough again) that varies by at most on , and the claim follows.

This gives the following corollary (also given as Exercise 16 in this previous blog post):

Corollary 5 (Weyl exponential sum estimate, inverse form II)Let be a discrete interval for some , and let polynomial of some degree at most for some . Iffor some , then there is a natural number such that for all .

One can obtain much better exponents here using Vinogradov’s mean value theorem; see Theorem 1.6 this paper of Wooley. (Thanks to Mariusz Mirek for this reference.) However, this weaker result already suffices for many applications, and does not need any result as deep as the mean value theorem.

*Proof:* To simplify notation we allow implied constants to depend on . As before, we may assume that for some small constant depending only on . We may also assume that for some large , as the claim is trivial otherwise (set ).

Applying Proposition 4, we can find a natural number and an arithmetic subprogression of such that and such that varies by at most on . Writing for some interval of length and some , we conclude that the polynomial varies by at most on . Taking order differences, we conclude that the coefficient of this polynomial is ; by the binomial theorem, this implies that differs by at most on from a polynomial of degree at most . Iterating this, we conclude that the coefficient of is for , and the claim then follows by inverting the change of variables (and replacing with a larger quantity such as as necessary).

For future reference we also record a higher degree version of the Vinogradov lemma.

Lemma 6 (Polynomial Vinogradov lemma)Let be a discrete interval for some , and let be a polynomial of degree at most for some such that for at least values of , for some . Then either

or else there is a natural number such that

for all .

*Proof:* We induct on . For this follows from Lemma 3 (noting that if then ), so suppose that and that the claim is already proven for . We now allow all implied constants to depend on .

For each , let denote the number of such that . By hypothesis, , and clearly , so we must have for choices of . For each such , we then have for choices of , so by induction hypothesis, either (5) or (6) holds, or else for choices of , there is a natural number such that

for , where are the coefficients of the degree polynomial . We may of course assume it is the latter which holds. By the pigeonhole principle we may take to be independent of .

Since , we have

for choices of , so by Lemma 3, either (5) or (6) holds, or else (after increasing as necessary) we have

We can again assume it is the latter that holds. This implies that modulo , so that

for choices of . Arguing as before and iterating, we obtain the claim.

The above results also extend to higher dimensions. Here is the higher dimensional version of Proposition 4:

Proposition 7 (Multidimensional Weyl exponential sum estimate, inverse form)Let and , and let be arithmetic progressions of length at most for each . Let be a polynomial of degrees at most in each of the variables separately. Iffor some , then there exists a subprogression of with for each such that varies by at most on .

A much more general statement, in which the polynomial phase is replaced by a nilsequence, and in which one does not necessarily assume the exponential sum is small, is given in Theorem 8.6 of this paper of Ben Green and myself, but it involves far more notation to even state properly.

*Proof:* We induct on . The case was established in Proposition 5, so we assume that and that the claim has already been proven for . To simplify notation we allow all implied constants to depend on . We may assume that for some small depending only on .

By a linear change of variables, we may assume that for all .

We write . First suppose that . Then by the pigeonhole principle we can find such that

and the claim then follows from the induction hypothesis. Thus we may assume that for some large depending only on . Similarly we may assume that for all .

By the triangle inequality, we have

The inner sum is , and the outer sum has terms. Thus, for choices of , one has

for some polynomials of degrees at most in the variables . For each obeying (7), we apply Corollary 5 to conclude that there exists a natural number such that

for (the claim also holds for but we discard it as being trivial). By the pigeonhole principle, there thus exists a natural number such that

for all and for choices of . If we write

where is a polynomial of degrees at most , then for choices of we then have

Applying Lemma 6 in the and the largeness hypotheses on the (and also the assumption that ) we conclude (after enlarging as necessary, and pigeonholing to keep independent of ) that

for all (note that we now include that case, which is no longer trivial) and for choices of . Iterating this, we eventually conclude (after enlarging as necessary) that

whenever for , with nonzero. Permuting the indices, and observing that the claim is trivial for , we in fact obtain (8) for all , at which point the claim easily follows by taking for each .

An inspection of the proof of the above result (or alternatively, by combining the above result again with many applications of Lemma 6) reveals the following general form of Proposition 4, which was posed as Exercise 17 in this previous blog post, but had a slight misprint in it (it did not properly treat the possibility that some of the could be small) and was a bit trickier to prove than anticipated (in fact, the reason for this post was that I was asked to supply a more detailed solution for this exercise):

Proposition 8 (Multidimensional Weyl exponential sum estimate, inverse form, II)Let be an natural number, and for each , let be a discrete interval for some . Letbe a polynomial in variables of multidegrees for some . If

for some , or else there is a natural number such that

Again, the factor of is natural in this bound. In the case, the option (10) may be deleted since (11) trivially holds in this case, but this simplification is no longer available for since one needs (10) to hold for *all* (not just one ) to make (11) completely trivial. Indeed, the above proposition fails for if one removes (10) completely, as can be seen for instance by inspecting the exponential sum , which has size comparable to regardless of how irrational is.

In Notes 5, we saw that the Gowers uniformity norms on vector spaces in high characteristic were controlled by classical polynomial phases .

Now we study the analogous situation on cyclic groups . Here, there is an unexpected surprise: the polynomial phases (classical or otherwise) are no longer sufficient to control the Gowers norms once exceeds . To resolve this problem, one must enlarge the space of polynomials to a larger class. It turns out that there are at least three closely related options for this class: the *local polynomials*, the *bracket polynomials*, and the *nilsequences*. Each of the three classes has its own strengths and weaknesses, but in my opinion the nilsequences seem to be the most natural class, due to the rich algebraic and dynamical structure coming from the nilpotent Lie group undergirding such sequences. For reasons of space we shall focus primarily on the nilsequence viewpoint here.

Traditionally, nilsequences have been defined in terms of linear orbits on nilmanifolds ; however, in recent years it has been realised that it is convenient for technical reasons (particularly for the quantitative “single-scale” theory) to generalise this setup to that of *polynomial* orbits , and this is the perspective we will take here.

A polynomial phase on a finite abelian group is formed by starting with a polynomial to the unit circle, and then composing it with the exponential function . To create a nilsequence , we generalise this construction by starting with a polynomial into a *nilmanifold* , and then composing this with a Lipschitz function . (The Lipschitz regularity class is convenient for minor technical reasons, but one could also use other regularity classes here if desired.) These classes of sequences certainly include the polynomial phases, but are somewhat more general; for instance, they *almost* include *bracket polynomial* phases such as . (The “almost” here is because the relevant functions involved are only piecewise Lipschitz rather than Lipschitz, but this is primarily a technical issue and one should view bracket polynomial phases as “morally” being nilsequences.)

In these notes we set out the basic theory for these nilsequences, including their equidistribution theory (which generalises the equidistribution theory of polynomial flows on tori from Notes 1) and show that they are indeed obstructions to the Gowers norm being small. This leads to the *inverse conjecture for the Gowers norms* that shows that the Gowers norms on cyclic groups are indeed controlled by these sequences.

In the previous lectures, we have focused mostly on the equidistribution or linear patterns on a subset of the integers , and in particular on intervals . The integers are of course a very important domain to study in additive combinatorics; but there are also other fundamental model examples of domains to study. One of these is that of a vector space over a finite field of prime order. Such domains are of interest in computer science (particularly when ) and also in number theory; but they also serve as an important simplified “dyadic model” for the integers. See this survey article of Green for further discussion of this point.

The additive combinatorics of the integers , and of vector spaces over finite fields, are analogous, but not quite identical. For instance, the analogue of an arithmetic progression in is a subspace of . In many cases, the finite field theory is a little bit simpler than the integer theory; for instance, subspaces are closed under addition, whereas arithmetic progressions are only “almost” closed under addition in various senses. (For instance, is closed under addition approximately half of the time.) However, there are some ways in which the integers are better behaved. For instance, because the integers can be generated by a single generator, a homomorphism from to some other group can be described by a single group element : . However, to specify a homomorphism from a vector space to one would need to specify one group element for each dimension of . Thus we see that there is a tradeoff when passing from (or ) to a vector space model; one gains a bounded torsion property, at the expense of conceding the bounded generation property. (Of course, if one wants to deal with arbitrarily large domains, one has to concede one or the other; the only additive groups that have both bounded torsion and boundedly many generators, are bounded.)

The starting point for this course (Notes 1) was the study of equidistribution of polynomials from the integers to the unit circle. We now turn to the parallel theory of equidistribution of polynomials from vector spaces over finite fields to the unit circle. Actually, for simplicity we will mostly focus on the *classical* case, when the polynomials in fact take values in the roots of unity (where is the characteristic of the field ). As it turns out, the non-classical case is also of importance (particularly in low characteristic), but the theory is more difficult; see these notes for some further discussion.

(Linear) Fourier analysis can be viewed as a tool to study an arbitrary function on (say) the integers , by looking at how such a function correlates with *linear phases* such as , where is the fundamental character, and is a frequency. These correlations control a number of expressions relating to , such as the expected behaviour of on arithmetic progressions of length three.

In this course we will be studying higher-order correlations, such as the correlation of with quadratic phases such as , as these will control the expected behaviour of on more complex patterns, such as arithmetic progressions of length four. In order to do this, we must first understand the behaviour of exponential sums such as

Such sums are closely related to the distribution of expressions such as in the unit circle , as varies from to . More generally, one is interested in the distribution of polynomials of one or more variables taking values in a torus ; for instance, one might be interested in the distribution of the quadruplet as both vary from to . Roughly speaking, once we understand these types of distributions, then the general machinery of quadratic Fourier analysis will then allow us to understand the distribution of the quadruplet for more general classes of functions ; this can lead for instance to an understanding of the distribution of arithmetic progressions of length in the primes, if is somehow related to the primes.

More generally, to find arithmetic progressions such as in a set , it would suffice to understand the equidistribution of the quadruplet in as and vary. This is the starting point for the fundamental connection between *combinatorics* (and more specifically, the task of finding patterns inside sets) and *dynamics* (and more specifically, the theory of equidistribution and recurrence in measure-preserving dynamical systems, which is a subfield of ergodic theory). This connection was explored in one of my previous classes; it will also be important in this course (particularly as a source of motivation), but the primary focus will be on finitary, and Fourier-based, methods.

The theory of equidistribution of polynomial orbits was developed in the linear case by Dirichlet and Kronecker, and in the polynomial case by Weyl. There are two regimes of interest; the (qualitative) *asymptotic regime* in which the scale parameter is sent to infinity, and the (quantitative) *single-scale regime* in which is kept fixed (but large). Traditionally, it is the asymptotic regime which is studied, which connects the subject to other asymptotic fields of mathematics, such as dynamical systems and ergodic theory. However, for many applications (such as the study of the primes), it is the single-scale regime which is of greater importance. The two regimes are not directly equivalent, but are closely related: the single-scale theory can be usually used to derive analogous results in the asymptotic regime, and conversely the arguments in the asymptotic regime can serve as a simplified model to show the way to proceed in the single-scale regime. The analogy between the two can be made tighter by introducing the (qualitative) *ultralimit regime*, which is formally equivalent to the single-scale regime (except for the fact that explicitly quantitative bounds are abandoned in the ultralimit), but resembles the asymptotic regime quite closely.

We will view the equidistribution theory of polynomial orbits as a special case of Ratner’s theorem, which we will study in more generality later in this course.

For the finitary portion of the course, we will be using *asymptotic notation*: , , or denotes the bound for some absolute constant , and if we need to depend on additional parameters then we will indicate this by subscripts, e.g. means that for some depending only on . In the ultralimit theory we will use an analogue of asymptotic notation, which we will review later in these notes.

Today, Prof. Margulis continued his lecture series, focusing on two specific examples of homogeneous dynamics applications to number theory, namely counting lattice points on algebraic varieties, and quantitative versions of the Oppenheim conjecture. (Due to lack of time, the third application mentioned in the previous lecture, namely metric theory of Diophantine approximation, was not covered.)

The final distinguished lecture series for the academic year here at UCLA is being given this week by Gregory Margulis, who is giving three lectures on “homogeneous dynamics and number theory”. In his first lecture, Prof. Margulis surveyed some classical problems in number theory that turn out, rather surprisingly, to have more or less equivalent counterparts in homogeneous dynamics – the theory of dynamical systems on homogeneous spaces .

As usual, any errors in this post are due to my transcription of the talk.

This week I was in Columbus, Ohio, attending a conference on equidistribution on manifolds. I talked about my recent paper with Ben Green on the quantitative behaviour of polynomial sequences in nilmanifolds, which I have blogged about previously. During my talk (and inspired by the immediately preceding talk of Vitaly Bergelson), I stated explicitly for the first time a generalisation of the van der Corput trick which morally underlies our paper, though it is somewhat buried there as we specialised it to our application at hand (and also had to deal with various quantitative issues that made the presentation more complicated). After the talk, several people asked me for a more precise statement of this trick, so I am presenting it here, and as an application reproving an old theorem of Leon Green that gives a necessary and sufficient condition as to whether a linear sequence on a nilmanifold is equidistributed, which generalises the famous theorem of Weyl on equidistribution of polynomials.

UPDATE, Feb 2013: It has been pointed out to me by Pavel Zorin that this argument does not fully recover the theorem of Leon Green; to cover all cases, one needs the more complicated van der Corput argument in our paper.

Ben Green and I have just uploaded our joint paper, “The distribution of polynomials over finite fields, with applications to the Gowers norms“, to the arXiv, and submitted to Contributions to Discrete Mathematics. This paper, which we first announced at the recent FOCS meeting, and then gave an update on two weeks ago on this blog, is now in final form. It is being made available simultaneously with a closely related paper of Lovett, Meshulam, and Samorodnitsky.

In the previous post on this topic, I focused on the negative results in the paper, and in particular the fact that the inverse conjecture for the Gowers norm fails for certain degrees in low characteristic. Today, I’d like to focus instead on the positive results, which assert that for polynomials in many variables over finite fields whose degree is less than the characteristic of the field, one has a satisfactory theory for the distribution of these polynomials. Very roughly speaking, the main technical results are:

- A
*regularity lemma*: Any polynomial can be expressed as a combination of a bounded number of other polynomials which are*regular*, in the sense that no non-trivial linear combination of these polynomials can be expressed efficiently in terms of lower degree polynomials. - A
*counting lemma*: A regular collection of polynomials behaves as if the polynomials were selected randomly. In particular, the polynomials are jointly equidistributed.

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