Capturing Forms in Dense Subsets of Finite Fields

TL;DR

This paper investigates the existence of certain algebraic configurations within dense subsets of finite fields, extending classical arithmetic problems into the finite field setting and providing size estimates for subsets containing these configurations.

Contribution

It introduces finite field analogues of an open problem in arithmetic Ramsey theory and offers estimates on subset sizes needed to contain specific algebraic forms.

Findings

Provides bounds on subset sizes in finite fields for containing linear and quadratic forms

Extends classical arithmetic Ramsey problems to finite field context

Offers new techniques for estimating configurations in finite algebraic structures

Abstract

An open problem of arithmetic Ramsey theory asks if given a finite $r$-colouring $c:\mathbb{N}\to\{1,...,r\}$ of the natural numbers, there exist $x,y\in \mathbb{N}$ such that $c(xy)=c(x+y)$ apart from the trivial solution $x=y=2$. More generally, one could replace $x+y$ with a binary linear form and $xy$ with a binary quadratic form. In this paper we examine the analogous problem in a finite field $\mathbb{F}_q$. Specifically, given a linear form $L$ and a quadratic from $Q$ in two variables, we provide estimates on the necessary size of $A\subset \mathbb{F}_q$ to guarantee that $L(x,y)$ and $Q(x,y)$ are elements of $A$ for some $x,y\in\mathbb{F}_q$.