An arithmetic analogue of Fox’s triangle removal argument

Abstract

We give an arithmetic version of the recent proof of the triangle removal lemma by Fox [Fox11], for the group \( \mathbb{F}_2^n \). A triangle in \( \mathbb{F}_2^n \) is a triple \( (x, y, z) \) such that \( x + y + z = 0 \). The triangle removal lemma for \( \mathbb{F}_2^n \) states that for every \( \varepsilon > 0 \) there is a \( \delta > 0 \), such that if a subset \( A \) of \( \mathbb{F}_2^n \) requires the removal of at least \( \varepsilon \cdot 2^n \) elements to make it triangle-free, then it must contain at least \( \delta \cdot 2^{2n} \) triangles.

This problem was first studied by Green [Gre05] who proved a lower bound on \( \delta \) using an arithmetic regularity lemma. Regularity-based lower bounds for triangle removal in graphs were recently improved by Fox, and we give a direct proof of an analogous improvement for triangle removal in \( \mathbb{F}_2^n \).

The improved lower bound was already known to follow (for triangle-removal in all groups) using Fox’s removal lemma for directed cycles and a reduction by Král, Serra, and Vena~\cite{KSV09} (see [Fox11, CF13]). The purpose of this note is to provide a direct Fourier-analytic proof for the group \( \mathbb{F}_2^n \).