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 $(g(n)\Gamma)_{n \in {\Bbb N}}$.

Before we extend to nilmanifolds, let us briefly review what happens for higher-dimensional torii ${\Bbb R}^m/{\Bbb Z}^m$. From the theory of Weyl sums, one can show that an infinite polynomial sequence $(P(n) \hbox{ mod } {\Bbb Z}^m)_{n \in {\Bbb N}}$ in a torus is either equidistributed, or is contained in a finite union of proper subtorii (cf. Kronecker’s theorem in the case when P is linear). Iterating this, we get a Ratner-type theorem for the torus, namely that every infinite polynomial sequence in a torus is equidistributed within a finite union of subtorii.

It turns out that a similar Ratner-type result holds for nilmanifolds, and is due to Leibman (with some earlier results in this direction by Leon Green, by Parry, and by Shah). Recall that a nilmanifold is a quotient space $G/\Gamma$, where G is a nilpotent Lie group (which for simplicity we shall take to be connected and simply connected, although these restrictions can be removed with a bit of effort), and $\Gamma$ is a discrete cocompact subgroup. A good example is the Heisenberg nilmanifold

$\displaystyle G/\Gamma := \begin{pmatrix} 1 & {\Bbb R} & {\Bbb R} \\ 0 & 1 & {\Bbb R} \\ 0 & 0 & 1 \end{pmatrix} /\begin{pmatrix} 1 & {\Bbb Z} & {\Bbb Z} \\ 0 & 1 & {\Bbb Z} \\ 0 & 0 & 1 \end{pmatrix}.$

There is a well-defined notion of a polynomial sequence in G – a sequence $g(n)$ which becomes trivial after finitely many applications of the differentiation operator $\partial$, defined as $\partial g(n) := g(n+1) g(n)^{-1}$. Leibman showed that an infinite polynomial sequence $(g(n)\Gamma)_{n \in {\Bbb N}}$ is either equidistributed in the nilmanifold $G/\Gamma$, or is else contained in a finite union of proper subnilmanifolds; iterating this, one can conclude that every infinite polynomial sequence is equidistributed in a finite union of nilmanifolds. One can also phrase this as a factorisation theorem: every polynomial sequence is the product of a constant sequence, a sequence equidistributed in a nilmanifold, and a periodic (and rational) sequence.

A typical application of Leibman’s theorem would be the assertion that for any real numbers $\alpha, \beta$, the sequence $( \lfloor \alpha n \rfloor \beta n )_{n \in {\Bbb N}}$ is either periodic, or equidistributed modulo 1, where $\lfloor x \rfloor$ is the integer part of x (except in some rare but explicitly describable cases, when $\alpha, \beta$ lie in a quadratic extension of ${\Bbb Q}$, in which one has a different distribution; see comments). I do not know of a way to prove this result which does not basically require one to prove a large part of Leibman’s theorem.

Leibman first proves his theorem in the linear case by heavy reliance on the ergodic theorem, an induction on the step of the nilmanifold, and some Fourier analysis (or representation theory) to handle the “vertical” behaviour of the nilmanifold, using some arguments of Parry. He then passes to the polynomial case by using a lifting trick of Furstenberg to linearise the sequence.

Extensions of these results beyond nilmanifolds, and in particular to finite volume homogeneous spaces, are certainly of interest (see for instance this article of Margulis) but are considerably more difficult, as one loses the ability to work by induction from the abelian case.

Now we turn to quantitative versions of the above statements, in which we look at the equidistribution properties of a finite polynomial sequence $(g(n) \Gamma)_{1 \leq n \leq N}$. One can define the notion of a finite sequence being equidistributed up to some error $\delta$, which roughly means that irregularities in the sequence can only be detected at scales $O(\delta)$ or below. Our first main result is that a finite polynomial sequence is either equidistributed up to error $\delta$, or is else concentrated within $O(\delta)$ of $O(\delta^{-O(1)})$ proper subnilmanifolds, whose “slopes” have height $O(\delta^{-O(1)})$. One can iterate this and conclude a Ratner-type theorem, namely that every finite polynomial sequence is within $O(\delta)$ of being equidistributed on a polynomial number of proper subnilmanifolds, whose slopes also have polynomially bounded heights. These polynomial bounds turn out to be important for our number theoretic applications, which will be discussed in a subsequent post. There is also a factorisation theorem, which asserts that every polynomial sequence can be expressed as the product of a slowly varying sequence, a sequence which is equidistributed in a nilmanifold, and a periodic rational sequence, with quantitative polynomial bounds on all of these assertions.

To illustrate our result with a concrete instance, we can assert that for any real numbers $\alpha, \beta$, any error tolerance $\delta > 0$, and any large integer, one can subdivide $\{1,\ldots,N\}$ into arithmetic progressions, each of density $\gg \delta^{O(1)}$, such that $\lfloor \alpha n \rfloor \beta n$ is either within $\delta$ of a constant, or is $\delta$-equidistributed, or is the push-forward of a $\delta$-equidistributed sequence by a quadratic polynomial with coefficients $O(\delta^{O(1)})$ (this latter case is technical, and rather rare, having to do with special subnilmanifolds of the Heisenberg nilmanifold).

One of the key difficulties in working in the quantitative setting (especially when one is insisting on polynomial bounds everywhere, and when N is fixed in advance) is that one can no longer use the ergodic theorem (unless one has quantitative control on the ergodicity, such as spectral gaps, which appear to be unavailable in this setting). Because of this, we were eventually forced to find a different approach than Leibman’s to these problems, and ended up with one based primarily on Fourier analysis in the vertical direction combined with the van der Corput inequality (very much in the spirit of the Weyl’s theory of equidistribution and exponential sums). Another new feature of the quantitative setting is the presence of error terms: a sequence may not lie exactly in a subnilmanifold, but instead deviates from it by a small error. This error can be eliminated, but often at the cost of increasing the “degree” (or other measures of “complexity”) of the polynomial sequence $g(n)$. This makes it far from obvious that iterative arguments guaranteed to terminate; it also seems to prevent one from using the Furstenberg lifting trick to linearise a sequence in the middle of an iteration argument, and so we were forced to work directly with polynomial sequences. For similar reasons, we had to perform a fair amount of algebraic computation to set up a good notion of the “complexity” of a polynomial sequence, which decreased with every stage of the main iteration argument. (In the end, this complexity is modeled by three parameters: the step s of the nilmanifold, the “commutator degree” D of the sequence g(n), and the “nonlinearity dimension” of the coefficient spaces for g(n), thus leading to a triple induction on these parameters in order to conclude the argument.)

[Update, Sep 26: inaccuracies with the $\alpha n \lfloor \beta n \rfloor$ example fixed; thanks to Ben Green for the corrections.]