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 n¹⁻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 ε n contains at least poly(ε )· nᵏ⁻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.