The Furstenberg-S\'ark\"ozy Theorem and Asymptotic Total Ergodicity Phenomena in Modular Rings

TL;DR

This paper extends ergodic theory concepts to modular rings, establishing conditions under which asymptotic total ergodicity occurs and deriving combinatorial results related to polynomial images and sumsets.

Contribution

It introduces the notion of asymptotic total ergodicity in modular rings and characterizes it via the growth of the least prime factor of the modulus, with applications to sumset decompositions.

Findings

Asymptotic total ergodicity occurs if and only if the least prime factor tends to infinity.

Large prime factors ensure sumset decompositions involving polynomial images.

Results have implications for combinatorial structures in modular arithmetic.

Abstract

The Furstenberg-S\'ark\"ozy 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 Poincar\'e 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 $|A||B| \geq \delta N^2$, then $\mathbb{Z}/N\mathbb{Z} = A + B + S$.