  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 $\mathbb{Z}/N\mathbb{Z} = A + B + S$.
