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arXiv · 2405.09402

A Generalization of Varnavides's Theorem

Asaf Shapira
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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.

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Asaf Shapira. 2024-05-15. A Generalization of Varnavides's Theorem. https://arxiv.org/abs/2405.09402

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