Revisiting sums and products in countable and finite fields

TL;DR

This paper develops new ergodic theorems for affine group actions over countable and finite fields, demonstrating the presence of complex polynomial and multiplicative patterns in large sets and colorings.

Contribution

It introduces a polynomial ergodic theorem for affine group actions, extends density results to finite fields, and develops a finitistic coloring method for pattern detection.

Findings

Large sets in characteristic zero fields contain polynomial and multiplicative patterns.

Finite field colorings of large enough size contain monochromatic polynomial and multiplicative patterns.

A new finitistic coloring trick enables elementary generalizations of known combinatorial theorems.

Abstract

We establish a polynomial ergodic theorem for actions of the affine group of a countable field $K$. As an application, we deduce--via a variant of Furstenberg's correspondence principle--that for fields of characteristic zero, any "large" set $E\subset K$ contains "many" patterns of the form $\{p(x)+y,xy\}$, for every non-constant polynomial $p(x)\in K[x]$. Our methods are flexible enough that they allow us to recover analogous density results in the setting of finite fields and, with the aid of a new finitistic variant of Bergelson's "colouring trick", show that for $r\in \mathbb{N}$ fixed, any $r-$colouring of a large enough finite field will contain monochromatic patterns of the form $\{x,p(x)+y,xy\}$. In a different direction, we obtain a double ergodic theorem for actions of the affine group of a countable field. An adaptation of the argument for affine actions of finite fields leads to a generalisation of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic "colouring trick", we provide a conditional, elementary generalisation of Green and Sanders' $\{x,y,x+y,xy\}$ theorem.