An Algebraic Hypergraph Regularity Lemma

TL;DR

This paper extends Tao's algebraic regularity lemma from graphs to hypergraphs and other definable sets in finite fields, providing new insights and answering open questions in algebraic combinatorics.

Contribution

It proves an algebraic hypergraph regularity lemma for definable sets in finite fields and extends it to difference fields, offering a new geometric perspective.

Findings

Established an algebraic hypergraph regularity lemma for finite fields.

Extended the lemma to definable sets in difference fields.

Provided a new geometric interpretation of the algebraic regularity lemma.

Abstract

Szemer\'edi's regularity lemma is a powerful tool in graph theory. It states that for every large enough graph, there exists a partition of the edge set with bounded size such that most induced subgraphs are quasirandom. When the graph is a definable set $\phi(x, y)$ in a finite field $F_q$, Tao's algebraic graph regularity lemma shows that there is a partition of the graph $\phi(x, y)$ such that all induced subgraphs are quasirandom and the error bound on quasirandomness is $O(q^{-1/4})$. In this work we prove an algebraic hypergraph regularity lemma for definable sets in finite fields, thus answering a question of Tao. We also extend the algebraic regularity lemma to definable sets in the difference fields $(F_q^{alg}, x^q)$ and we offer a new point of view on the geometric content of the algebraic regularity lemma.