English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We obtain new combinatorial results about polynomial configurations in largesubsets of finite fields and rings by utilizing the phenomenon of asymptotictotal ergodicity (previously studied for actions of $\mathbb{Z}$ on modularrings $\mathbb{Z}/N\mathbb{Z}$ in [Bergelson--Best, 2023]) in the context ofactions of the polynomial ring $\mathbb{F}[t]$ over a finite field$\mathbb{F}$. Drawing inspiration from the well-understood limiting behavior ofpolynomial ergodic averages in totally ergodic systems, we show that thenatural action of $\mathbb{F}[t]$ on a sequence of quotient rings$\mathbb{F}[t]/Q_n(t)\mathbb{F}[t]$, $Q_n(t) \in \mathbb{F}[t]$, isasymptotically totally ergodic if and only if every polynomial $P(x) \in(\mathbb{F}[t])[x]$ with $P(0) = 0$ asymptotically equidistributes in asubgroup of $\mathbb{F}[t]/Q_n(t)\mathbb{F}[t]$. We then derive severalcombinatorial consequences:  (1) We establish a power saving bound for the Furstenberg--S\'ark\"ozytheorem over finite fields of fixed characteristic, complementing recent workof Li and Sauermann giving power saving bounds using the polynomial method.  (2) We prove an enhancement of the Furstenberg--S\'ark\"ozy theoremguaranteeing many pairs $(x,y)$ with $x \in A$ and $x + P(y) \in B$ whenever$A$ and $B$ are large subsets of a quotient ring$\mathbb{F}[t]/Q(t)\mathbb{F}[t]$ that exhibits a sufficiently high level ofapproximate total ergodicity and the polynomial $P$ satisfies a rather generalcondition related to equidistributional properties studied in[Bergelson--Leibman, 2016]. We also show that, in the absence of asymptotictotal ergodicity and an equidistribution condition on P, one cannot hope forsuch a refinement of the Furstenberg--S\'ark\"ozy theorem.  (3) We produce new families of examples of partition regular polynomialequations over finite fields.

Asymptotic total ergodicity for actions of $\mathbb{F}[t]$ and polynomial configurations over finite fields and rings