A generalisation 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 n^1−c contains a solution of E in distinct integers. The problem of characterising the sparse equations, first raised by Ruzsa in the 90s, 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 εn contains at least poly(ε) · n^k−1 solutions of E. It is clear that every abundant E is sparse, and Girão, 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 four variables. We further discuss a generalisation of this problem which applies to all linear equations.