## Abstract

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.

Original language | English |
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Journal | Combinatorics Probability and Computing |

DOIs | |

State | Accepted/In press - 2024 |

## Keywords

- Roth
- Ruzsa
- supersaturation
- Varnavides

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics