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A generalisation of Varnavides’s theorem
- Part of
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- Journal:
- Combinatorics, Probability and Computing / Volume 33 / Issue 6 / November 2024
- Published online by Cambridge University Press:
- 29 May 2024, pp. 724-728
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A linear equation
$E$ is said to be sparse if there is 
$c\gt 0$ 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 
$\varepsilon n$ contains at least 
$\text{poly}(\varepsilon )\cdot n^{k-1}$ 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.
