A Generalization of Varnavides's Theorem

TL;DR

This paper proves that for certain 4-variable linear equations, being sparse is equivalent to being abundant, advancing the understanding of solution-rich subsets in additive combinatorics.

Contribution

It establishes the equivalence between sparsity and abundance for 4-variable linear equations, extending the characterization of such equations in additive combinatorics.

Findings

Proves the equivalence for 4-variable equations

Extends the characterization of sparse and abundant equations

Discusses a generalization to all linear equations

Abstract

A linear equation $E$ is said to be sparse if there is $c>0$ so that every subset of $[n]$ of size $n^{1-c}$ contains a solution of $E$ in distinct integers. The problem of characterizing the sparse equations, first raised by Ruzsa in the 90's, is one of the most important open problems in additive combinatorics. We say that $E$ in $k$ variables is abundant if every subset of $[n]$ of size $\varepsilon n$ contains at least poly$(\varepsilon)\cdot n^{k-1}$ solutions of $E$. It is clear that every abundant $E$ is sparse, and Gir\~{a}o, Hurley, Illingworth and Michel asked if the converse implication also holds. In this note we show that this is the case for every $E$ in $4$ variables. We further discuss a generalization of this problem which applies to all linear equations.