Ergodic Theory and Dynamical Systems SeminarㅣOn the polynomial Szemerédi theorem over finite commutative rings

Abstract: The polynomial Szemerédi theorem implies that, for any $\delta \in (0,1)$, any family $\{P_1,\ldots, P_m\} \subset \Z[y]$ of nonconstant polynomials with constant term zero, and any sufficiently large $N \in \N$, every subset of $\{1,\ldots, N\}$ of cardinality at least $\delta N$ contains a nontrivial configuration of the form $\{x,x+P_1(y),\ldots, x+P_m(y)\}$. When the polynomials are assumed independent, one can expect a sharper result to hold over finite fields, special cases of which were proven recently in various articles by Bourgain and others, culminating with a 2018 result of Peluse, which deals with the general case of independent polynomials. In this talk we discuss, over general finite commutative rings, a version of the polynomial Szemerédi theorem for multivariable independent polynomials $\{P_1,\ldots, P_m\} \subset \Z[y_1,\ldots, y_n]$, deriving new combinatorial consequences, such as the following. Let $\mathcal R$ be a collection of finite commutative rings satisfying a technical condition which limits the amount of torsion. There exists $\gamma \in (0,1)$ such that, for every $R \in \mathcal R$, every subset $A \subset R$ of cardinality at least $|R|^{1-\gamma}$ contains a nontrivial configuration $\{x,x+P_1(y_1,\ldots,y_n),\ldots, x+P_m(y_1,\ldots, y_n)\}$ for some $x,y_1,\ldots, y_n \in R$. The move from finite fields to finite commutative rings introduces many unexpected obstacles to our ergodic arguments, which I will give a flavor of.