On a Ramsey--Tur\'{a}n variant of Roth's theorem

TL;DR

This paper establishes a Ramsey--Turán type result for Roth's theorem, linking the structure of solution-free sets in finite fields to the sum of coefficients in linear equations.

Contribution

It proves an equivalence between the size of structured solution-free sets and the zero-sum condition of coefficients in linear equations over finite fields.

Findings

Solution-free sets with small structured sum have size o(p)

Zero-sum subset of coefficients characterizes the structured solution-free sets

Generalizes Roth's theorem to structured set contexts

Abstract

A classical theorem of Roth states that the maximum size of a solution-free set of a homogeneous linear equation $\mathcal{L}$ in $\mathbb{F}_p$ is $o(p)$ if and only if the sum of the coefficients of $\mathcal{L}$ is $0$. In this paper, we prove a Ramsey--Tur\'{a}n variant of Roth's theorem, with respect to a natural notion of ``structured'' sets introduced by Erd\H{o}s and S\'ark\"ozy in the 1970's. Namely, we show that the following statements are equivalent: $(a)$ Every solution-free set $A$ of $\mathcal{L}$ in $\mathbb{F}_p$ with $\alpha(\mathrm{Cay}_{\mathbb{F}_p}(A)) = o(p)$ has size $o(p)$. $(b)$ There exists a non-empty \emph{subset} of coefficients of $\mathcal{L}$ with zero sum.