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Asymptotic total ergodicity for actions of $\mathbb{F}[t]$ and polynomial configurations over finite fields and rings by Ackelsberg, Ethan; Bergelson, Vitaly is a scholarly article available to read on EtoBox.

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We obtain new combinatorial results about polynomial configurations in large subsets of finite fields and rings by utilizing the phenomenon of asymptotic total ergodicity (previously studied for actions of $\mathbb{Z}$ on modular rings $\mathbb{Z}/N\mathbb{Z}$ in [Bergelson--Best, 2023]) in the context of actions of the polynomial ring $\mathbb{F}[t]$ over a finite field $\mathbb{F}$. Drawing inspiration from the well-understood limiting behavior of polynomial ergodic averages in totally ergodic systems, we show that the natural action of $\mathbb{F}[t]$ on a sequence of quotient rings $\mathbb{F}[t]/Q_n(t)\mathbb{F}[t]$, $Q_n(t) \in \mathbb{F}[t]$, is asymptotically totally ergodic if and only if every polynomial $P(x) \in (\mathbb{F}[t])[x]$ with $P(0) = 0$ asymptotically equidistributes in a subgroup of $\mathbb{F}[t]/Q_n(t)\mathbb{F}[t]$. We then derive several combinatorial consequences: (1) We establish a power saving bound for the Furstenberg--S\'ark\"ozy theorem over finite fields of fixed characteristic, complementing recent work of Li and Sauermann giving power saving bounds using the polynomial method. (2) We prove an enhancement of the Furstenberg--S\'ark\"ozy theorem g

Author

Ackelsberg, Ethan; Bergelson, Vitaly

Published

2023

Language

EN