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Note: this post is of a particularly technical nature, in particular presuming familiarity with nilsequences, nilsystems, characteristic factors, etc., and is primarily intended for experts.

As mentioned in the previous post, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms:

Theorem 1 (Inverse theorem for Gowers norms) Let ${N \geq 1}$ and ${s \geq 1}$ be integers, and let ${\delta > 0}$. Suppose that ${f: {\bf Z} \rightarrow [-1,1]}$ is a function supported on ${[N] := \{1,\dots,N\}}$ such that

$\displaystyle \frac{1}{N^{s+2}} \sum_{n,h_1,\dots,h_{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.$

Then there exists a filtered nilmanifold ${G/\Gamma}$ of degree ${\leq s}$ and complexity ${O_{s,\delta}(1)}$, a polynomial sequence ${g: {\bf Z} \rightarrow G}$, and a Lipschitz function ${F: G/\Gamma \rightarrow {\bf R}}$ of Lipschitz constant ${O_{s,\delta}(1)}$ such that

$\displaystyle \frac{1}{N} \sum_n f(n) F(g(n) \Gamma) \gg_{s,\delta} 1.$

This result was conjectured earlier by Ben Green and myself; this conjecture was strongly motivated by an analogous inverse theorem in ergodic theory by Host and Kra, which we formulate here in a form designed to resemble Theorem 1 as closely as possible:

Theorem 2 (Inverse theorem for Gowers-Host-Kra seminorms) Let ${s \geq 1}$ be an integer, and let ${(X, T)}$ be an ergodic, countably generated measure-preserving system. Suppose that one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{s+1}} \sum_{h_1,\dots,h_{s+1} \in [N]} \int_X \prod_{\omega \in \{0,1\}^{s+1}} f(T^{\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}}x)\ d\mu(x)$

$\displaystyle > 0$

for all non-zero ${f \in L^\infty(X)}$ (all ${L^p}$ spaces are real-valued in this post). Then ${(X,T)}$ is an inverse limit (in the category of measure-preserving systems, up to almost everywhere equivalence) of ergodic degree ${\leq s}$ nilsystems, that is to say systems of the form ${(G/\Gamma, x \mapsto gx)}$ for some degree ${\leq s}$ filtered nilmanifold ${G/\Gamma}$ and a group element ${g \in G}$ that acts ergodically on ${G/\Gamma}$.

It is a natural question to ask if there is any logical relationship between the two theorems. In the finite field category, one can deduce the combinatorial inverse theorem from the ergodic inverse theorem by a variant of the Furstenberg correspondence principle, as worked out by Tamar Ziegler and myself, however in the current context of ${{\bf Z}}$-actions, the connection is less clear.

One can split Theorem 2 into two components:

Theorem 3 (Weak inverse theorem for Gowers-Host-Kra seminorms) Let ${s \geq 1}$ be an integer, and let ${(X, T)}$ be an ergodic, countably generated measure-preserving system. Suppose that one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{s+1}} \sum_{h_1,\dots,h_{s+1} \in [N]} \int_X \prod_{\omega \in \{0,1\}^{s+1}} T^{\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}} f\ d\mu$

$\displaystyle > 0$

for all non-zero ${f \in L^\infty(X)}$, where ${T^h f := f \circ T^h}$. Then ${(X,T)}$ is a factor of an inverse limit of ergodic degree ${\leq s}$ nilsystems.

Theorem 4 (Pro-nilsystems closed under factors) Let ${s \geq 1}$ be an integer. Then any factor of an inverse limit of ergodic degree ${\leq s}$ nilsystems, is again an inverse limit of ergodic degree ${\leq s}$ nilsystems.

Indeed, it is clear that Theorem 2 implies both Theorem 3 and Theorem 4, and conversely that the two latter theorems jointly imply the former. Theorem 4 is, in principle, purely a fact about nilsystems, and should have an independent proof, but this is not known; the only known proofs go through the full machinery needed to prove Theorem 2 (or the closely related theorem of Ziegler). (However, the fact that a factor of a nilsystem is again a nilsystem was established previously by Parry.)

The purpose of this post is to record a partial implication in reverse direction to the correspondence principle:

Proposition 5 Theorem 1 implies Theorem 3.

As mentioned at the start of the post, a fair amount of familiarity with the area is presumed here, and some routine steps will be presented with only a fairly brief explanation.

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 $(g^n x)_{n=1}^\infty$ on a nilmanifold $G/\Gamma$ 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.

The last two lectures of this course will be on Ratner’s theorems on equidistribution of orbits on homogeneous spaces. Due to lack of time, I will not be able to cover all the material here that I had originally planned; in particular, for an introduction to this family of results, and its connections with number theory, I will have to refer readers to my previous blog post on these theorems. In this course, I will discuss two special cases of Ratner-type theorems. In this lecture, I will talk about Ratner-type theorems for discrete actions (of the integers ${\Bbb Z})$ on nilmanifolds; this case is much simpler than the general case, because there is a simple criterion in the nilmanifold case to test whether any given orbit is equidistributed or not. Ben Green and I had need recently to develop quantitative versions of such theorems for a number-theoretic application. In the next and final lecture of this course, I will discuss Ratner-type theorems for actions of $SL_2({\Bbb R})$, which is simpler in a different way (due to the semisimplicity of $SL_2({\Bbb R})$, and lack of compact factors).

Ben Green and I have just uploaded our paper “The quantitative behaviour of polynomial orbits on nilmanifolds” to the arXiv (and shortly to be submitted to a journal, once a companion paper is finished). This paper grew out of our efforts to prove the Möbius and Nilsequences conjecture MN(s) from our earlier paper, which has applications to counting various linear patterns in primes (Dickson’s conjecture). These efforts were successful – as the companion paper will reveal – but it turned out that in order to establish this number-theoretic conjecture, we had to first establish a purely dynamical quantitative result about polynomial sequences in nilmanifolds, very much in the spirit of the celebrated theorems of Marina Ratner on unipotent flows; I plan to discuss her theorems in more detail in a followup post to this one.In this post I will not discuss the number-theoretic applications or the connections with Ratner’s theorem, and instead describe our result from a slightly different viewpoint, starting from some very simple examples and gradually moving to the general situation considered in our paper.

To begin with, consider a infinite linear sequence $(n \alpha + \beta)_{n \in {\Bbb N}}$ in the unit circle ${\Bbb R}/{\Bbb Z}$, where $\alpha, \beta \in {\Bbb R}/{\Bbb Z}$. (One can think of this sequence as the orbit of $\beta$ under the action of the shift operator $T: x \mapsto x +\alpha$ on the unit circle.) This sequence can do one of two things:

1. If $\alpha$ is rational, then the sequence $(n \alpha + \beta)_{n \in {\Bbb N}}$ is periodic and thus only takes on finitely many values.
2. If $\alpha$ is irrational, then the sequence $(n \alpha + \beta)_{n \in {\Bbb N}}$ is dense in ${\Bbb R}/{\Bbb Z}$. In fact, it is not just dense, it is equidistributed, or equivalently that

$\displaystyle\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N F( n \alpha + \beta ) = \int_{{\Bbb R}/{\Bbb Z}} F$

for all continuous functions $F: {\Bbb R}/{\Bbb Z} \to {\Bbb C}$. This statement is known as the equidistribution theorem.

We thus see that infinite linear sequences exhibit a sharp dichotomy in behaviour between periodicity and equidistribution; intermediate scenarios, such as concentration on a fractal set (such as a Cantor set), do not occur with linear sequences. This dichotomy between structure and randomness is in stark contrast to exponential sequences such as $( 2^n \alpha)_{n \in {\Bbb N}}$, which can exhibit an extremely wide spectrum of behaviours. For instance, the question of whether $(10^n \pi)_{n \in {\Bbb N}}$ is equidistributed mod 1 is an old unsolved problem, equivalent to asking whether $\pi$ is normal base 10.

Intermediate between linear sequences and exponential sequences are polynomial sequences $(P(n))_{n \in {\Bbb N}}$, where P is a polynomial with coefficients in ${\Bbb R}/{\Bbb Z}$. A famous theorem of Weyl asserts that infinite polynomial sequences enjoy the same dichotomy as their linear counterparts, namely that they are either periodic (which occurs when all non-constant coefficients are rational) or equidistributed (which occurs when at least one non-constant coefficient is irrational). Thus for instance the fractional parts $\{ \sqrt{2}n^2\}$ of $\sqrt{2} n^2$ are equidistributed modulo 1. This theorem is proven by Fourier analysis combined with non-trivial bounds on Weyl sums.

For our applications, we are interested in strengthening these results in two directions. Firstly, we wish to generalise from polynomial sequences in the circle ${\Bbb R}/{\Bbb Z}$ to polynomial sequences $(g(n)\Gamma)_{n \in {\Bbb N}}$ in other homogeneous spaces, in particular nilmanifolds. Secondly, we need quantitative equidistribution results for finite orbits $(g(n)\Gamma)_{1 \leq n \leq N}$ rather than qualitative equidistribution for infinite orbits $(g(n)\Gamma)_{n \in {\Bbb N}}$.