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

The Furstenberg-Sarkozy 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 $P(0)=0$. Furstenbergs approach relies on a correspondence principle and a polynomial version of the Poincare 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 $mathbb{Z}/Nmathbb{Z}$. We show that a sequence of modular rings $mathbb{Z}/N_mmathbb{Z}$, $m in mathbb{N},$ 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 $S = { Q(n) : n in mathbb{Z}/Nmathbb{Z}}$. For any integer $N > 1$ with $mathrm{lpf}(N)$ sufficiently large, if $A$ and $B$ are subsets of $mathbb{Z}/Nmathbb{Z}$ such that $|A||B| geq delta N^2$, then $mathbb{Z}/Nmathbb{Z} = A + B + S$.