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

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 $c(A):= \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N \mu(A \cap T^{-P(n)}A) > 0.$Thelimit$c(A)$willhaveitsoptimalvalueof$\mu(A)^2$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 $\mathbb{Z}\mathrm{/}N\mathbb{Z}$. We show that a sequence of modular rings $\mathbb{Z}\mathrm{/}{N}_m\mathbb{Z}$, $m\in \mathbb{N},$ is asymptotically totally ergodic if and only if $\mathrm{l}\mathrm{p}\mathrm{f}(N_m)$, 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$\mathbb{Z}$}$\mathrm{.}Foranyinteger$N > 1$with$$\mathrm{l}\mathrm{p}\mathrm{f}$(N)$sufficientlylarge,if$A$and$B$aresubsetsof$$\mathbb{Z}$/N$\mathbb{Z}$$ such that $\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }\ge \delta N^2$, then $\mathbb{Z}\mathrm{/}N\mathbb{Z}=A+B+S$.