The Furstenberg-Sárközy Theorem and Asymptotic Total Ergodicity Phenomena in Modular Rings

Abstract

The Furstenberg-Sárközy theorem asserts that the difference set $E-E$ of a subset $E\subset \mathbb{N}$ with positive upper density intersects the image set of any polynomial $P\in \mathbb{Z}[n]$ for which $P(0)=0$. Furstenberg's approach relies on a correspondence principle and a polynomial version of the Poincaré recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system $(X,\mathcal{B},\mu ,T)$ and set $A\in \mathcal{B}$ with $\mu (A)>0$, one has $c(A):={\lim }_{N\to \infty }\frac{1}{N}{\sum }_{n=1}^N\mu (A\cap T^{-P(n)}A)>0.$ The limit $c(A)$ will have its optimal value of $\mu (A{)}^2$ when $T$ is totally ergodic. Motivated by the possibility of new combinatorial applications, we define the notion of asymptotic total ergodicity in the setting of modular rings $\mathbb{Z}/N\mathbb{Z}$. We show that a sequence of modular rings $\mathbb{Z}/N_m\mathbb{Z}$, $m\in \mathbb{N},$ is asymptotically totally ergodic if and only if $\mathrm{lpf}(N_m)$, the least prime factor of $N_m$, grows to infinity. From this fact, we derive some combinatorial consequences, for example the following. Fix $\delta \in (0,1]$ and a (not necessarily intersective) polynomial $Q\in \mathbb{Q}[n]$ such that $Q(\mathbb{Z})\subseteq \mathbb{Z}$, and write $S=\{Q(n):n\in \mathbb{Z}/N\mathbb{Z}\}$. For any integer $N>1$ with $\mathrm{lpf}(N)$ sufficiently large, if $A$ and $B$ are subsets of $\mathbb{Z}/N\mathbb{Z}$ such that $|A||B|\ge \delta N^2$, then $\mathbb{Z}/N\mathbb{Z}=A+B+S$.

arXiv comment29 pages, update with referee's comments