A Generalization of Varnavides's Theorem

A linear equation E is said to be sparse if there is $c>0$ so that every subset of $[n]$ of size n¹⁻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 ε n contains at least poly(ε)· nᵏ⁻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.