On the Furstenberg-Katznelson constant for the IP Szemeredi theorem over finite fields

TL;DR

This paper investigates the analogues of Furstenberg-Katznelson constants for the IP Szemeredi theorem in finite fields, providing qualitative results and quantitative bounds using ergodic theory.

Contribution

It introduces and analyzes the finite field counterparts of these constants, extending ergodic-theoretic methods to vector spaces over finite fields.

Findings

Established qualitative results for the finite field constants.

Derived strong quantitative bounds in special cases like Roth's theorem.

Used ergodic theory to study characteristic factors and limits of multiple averages.

Abstract

Bergelson et al. observed that Furstenberg's proof of Szemeredi's theorem provides a positive lower bound on the density of arithmetic progressions in sets of positive density in the integers. Namely, for every $\delta\in(0,1]$ and every $k\in \mathbb{N}$, there exists a positive constant $c=c(k,\delta)>0$ such that $$\{n\in \mathbb{N} : d(E\cap (E-n)\cap\dots\cap (E-(k-1)n))>c(k,\delta)\} \neq \emptyset$$ whenever $d(E)\ge \delta$. Similarly, Furstenberg and Katznelson proved the IP Szemeredi theorem, establishing in particular the existence of a constant $c_{\mathrm{IP}}=c_{\mathrm{IP}}(k,\delta)>0$ such that $$\{n\in \mathbb{N} : d(E\cap (E-n)\cap\dots\cap (E-(k-1)n))>c_{\mathrm{IP}}(k,\delta)\}$$ is $\mathrm{IP}^*$ whenever $d(E)\ge \delta$. In this paper, we study analogues of $c$ and $c_{\mathrm{IP}}$ and their ergodic-theoretic counterparts, $c^{\mathrm{rec}}$ and $c_{\mathrm{IP}}^{\mathrm{rec}}$, for vector spaces over finite fields. We provide a qualitative result and in special cases such as Roth's theorem and the IP-Roth theorem, we also provide strong quantitative bounds for these constants. Our tools are primarily ergodic theoretic; we study the characteristic factors and limit of multiple ergodic averages along $\mathrm{IP}$s in vector spaces over finite fields.