**Title: **The Furstenberg-Sárközy theorem and asymptotic total ergodicity

**Speaker: **Andrew Best – Ohio State University

**Abstract: **The Furstenberg-Sárközy theorem asserts that the difference set E-E of a subset E of the natural numbers with positive upper density contains a (nonzero) square. Furstenberg’s approach relies on a correspondence principle and a version of the Poincaré recurrence theorem along squares; the latter is shown via the result that for any measure-preserving system $(X,\mathcal{B},\mu,T)$ and set A with positive measure, the ergodic average $\frac{1}{N} \sum_{n=1}^N \mu(A \cap T^{-n^2}A)$ has a positive limit c(A) as N tends to infinity. Motivated — by what? we shall see — to optimize the value of c(A), 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 (Z/N_m Z) is asymptotically totally ergodic if and only if the least prime factor of N_m grows to infinity. From this fact, we derive some combinatorial consequences. These results are based on joint work with Vitaly Bergelson.