A decomposition of multicorrelation sequences for commuting transformations along primes

A decomposition of multicorrelation sequences for commuting transformations along primes, Discrete Analysis 2021:4, 27 pp.

Szemerédi’s theorem asserts that for every positive integer $k$ and every $\delta>0$ there exists $n$ such that every subset of $\{1,2,\dots,n\}$ of size at least $\delta n$ contains an arithmetic progression of length $k$. It is not too hard to prove that the theorem is equivalent to the following statement.

Theorem. For every positive integer $k$ and every $\delta>0$ there is a constant $c=c(k,\delta)>0$ such that if $f:\mathbb Z_n\to[0,1]$ and $\mathbb E_xf(x)\geq\delta$, then
$$\mathbb E_{x,d}f(x)f(x+d)\dots f(x+(k-1)d)\geq c.$$

Here $\mathbb Z_n$ denotes the additive group of integers mod $n$ and $\mathbb E_{x,d}$ denotes the expectation when $x,d$ are chosen uniformly and independently at random from $\mathbb Z_n$.

To see the connection between the above theorem and Szemerédi’s theorem, note that if $f$ is the characteristic function of a set $A$ of size at least $\delta n$, then the conclusion of the theorem is that the number of pairs $(x,d)$ such that $x, x+d, \dots, x+(k-1)d$ all belong to $A$ is at least $cn^2$.

The Furstenberg correspondence principle is a general result that shows that many combinatorial statements are equivalent to statements about measure-preserving dynamical systems. In particular, it shows that Szemerédi’s theorem is equivalent to the following statement.

Theorem. Let $(X,\mathcal B,\mu)$ be a probability space, and let, $T:X\to X$ be a measure-preserving transformation (meaning that $\mu(T^{-1}(B))=\mu(B)$ for every $B\in\mathcal B$). Then for every positive integer $k$ and every set $A\in\mathcal B$ with $\mu(A)>0$ there exists a positive integer $n$ such that
$$\mu(A\cap T^{-n}A\cap T^{-2n}A\cap\dots\cap T^{-kn}A)>0.$$

Thus, if we define $\alpha(n)$ to be the measure of the intersection above, Szemerédi’s theorem for progressions of length $k+1$ is equivalent to the assertion that $\alpha$ is not identically zero.

The sequence $(\alpha(n))$ is an example of a multicorrelation sequence. It turns out to be important for many reasons not merely to prove that multicorrelation sequences do not vanish, but to obtain a more detailed understanding of their structure, a line of enquiry that was initiated by Bergelson, Host and Kra in a very influential paper of 2005. In particular, they showed that multicorrelation sequences, including more general sequences of the form
$$\int_X f_0.T^nf_1.^{2n}f_2\dots T^{kn}f_k d\mu,$$
are close to structured objects known as $k$-step nilsequences. Roughly speaking, a $k$-step nilsequence is obtained as follows. One takes a $k$-step nilpotent Lie group $G$, a cocompact subgroup $\Gamma$, an element $x\in G/\Gamma$, an element $g\in G$, and a continuous function $f:G/\Gamma\to\mathbb R$. The nilsequence is then the sequence $f(x), f(gx), f(g^2x),\dots$. (Here we are writing $gx$ as shorthand for the result of the obvious left action of $G$ on $G/\Gamma$.)

This result has been generalized in several directions, which reflect various different important generalizations of Szemerédi’s theorem. One is to look at sequences of the form
$$\alpha(n)=\int_X f_0.T_1^nf_1\dots T_k^nf_k d\mu,$$
where $T_1,\dots,T_k$ are commuting measure-preserving transformations. These are closely related to multidimensional versions of Szemerédi’s theorem. Another is to look at sequences of the form
$$\alpha(n)=\int_X f_0.T_1^{q_1(n)}f_1\dots T_k^{q_k(n)}f_k d\mu,$$
which relate to the polynomial Szemerédi theorem of Bergelson and Leibman.

Another direction of generalization is to consider special subsequences of the sequences above. For instance, for the original sequence and for the polynomial sequence, closeness to a nilsequence has been proved even if one restricts to a subsequence of the form $q(p)$, where $q$ is a non-constant polynomial and $p$ runs through the primes.

The main result of this paper solves a problem of Frantzikinakis by obtaining a simultaneous generalization of all the results above. The precise statement is Theorem B of their paper, but a special case, which contained the main difficulties, concerns the multidimensional multicorrelation sequences
$$\alpha(n)=\int f_0.T_1^nf_1\dots T_k^nf_k d\mu$$
mentioned above. Here they prove that there is a $k$-step nilsequence $\nu$ such that $\alpha$ is close (in a sense defined in the paper, but roughly speaking it means close on average) to $\nu$ both along the positive integers and along the primes. To do this, they draw on previous ideas, such as the “transference” method of Green and Tao, but the obvious way of applying those ideas runs into a significant difficulty, so in order to carry out the approach, the authors need additional ideas. In particular, they obtain further properties of the nilsequence approximations to $\alpha(n)$ above.