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In this lecture, we move away from recurrence, and instead focus on the structure of topological dynamical systems. One remarkable feature of this subject is that starting from fairly “soft” notions of structure, such as topological structure, one can extract much more “hard” or “rigid” notions of structure, such as geometric or algebraic structure. The key concept needed to capture this structure is that of an isometric system, or more generally an isometric extension, which we shall discuss in this lecture. As an application of this theory we characterise the distribution of polynomial sequences in torii (a baby case of a variant of Ratner’s theorem due to (Leon) Green, which we will cover later in this course).

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