A generalisation of Varnavides's theorem

A linear equation <![CDATA[ $E$ ]]> is said to be sparse if there is <![CDATA[ $c\gt 0$ ]]> so that every subset of <![CDATA[ $[n]$ ]]> of size <![CDATA[ $n^{1-c}$ ]]> contains a solution of <![CDATA[ $E$ ]]> 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 <![CDATA[ $E$ ]]> in <![CDATA[ $k$ ]]> variables is abundant if every subset of <![CDATA[ $[n]$ ]]> of size <![CDATA[ $\varepsilon n$ ]]> contains at least <![CDATA[ $\text{poly}(\varepsilon)\cdot n^{k-1}$ ]]> solutions of <![CDATA[ $E$ ]]>. It is clear that every abundant <![CDATA[ $E$ ]]> 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 <![CDATA[ $E$ ]]> in four variables. We further discuss a generalisation of this problem which applies to all linear equations.