A polynomial bound for the arithmetic $k$-cycle removal lemma in vector spaces

TL;DR

This paper establishes a polynomial bound for the arithmetic $k$-cycle removal lemma in vector spaces over finite fields, improving understanding of combinatorial properties relevant to property testing.

Contribution

It provides the first polynomial bounds for the lemma when $k>3$ in vector spaces over finite fields, extending previous results for $k=3$ with a new proof strategy.

Findings

Polynomial bounds for $k>3$ in $ ext{F}_p^n$

Extension of $k=3$ results to higher $k$

New proof strategy overcoming previous limitations

Abstract

For each $k\geq 3$, Green proved an arithmetic $k$-cycle removal lemma for any abelian group $G$. The best known bounds relating the parameters in the lemma for general $G$ are of tower-type. For $k>3$, even in the case $G=\mathbb{F}_2^n$ no better bounds were known prior to this paper. This special case has received considerable attention due to its close connection to property testing of boolean functions. For every $k\geq 3$, we prove a polynomial bound relating the parameters for $G=\mathbb{F}_p^n$, where $p$ is any fixed prime. This extends the result for $k=3$ by the first two authors. Due to substantial issues with generalizing the proof of the $k=3$ case, a new strategy is developed in order to prove the result for $k>3$.