Regularity and counting lemmas for multidimensional matrices

TL;DR

This paper extends regularity and counting lemmas from graph theory to multidimensional matrices, introducing new concepts to handle higher dimensions and providing foundational tools for analyzing complex matrix structures.

Contribution

It generalizes regularity and counting lemmas to multidimensional matrices, including the introduction of ε-regular patterns for higher-dimensional cases.

Findings

A multidimensional regularity lemma based on graph regularity translation.

An ε-regularity condition sufficient for 2D matrix counting lemmas.

Introduction of ε-regular patterns enabling multidimensional counting lemmas.

Abstract

In the present paper we propose generalizations of the regularity and counting lemmas for multidimensional matrices under a finite alphabet. Firstly, we prove a variant of a multidimensional regularity lemma with the help of a translation of $\varepsilon$-regularity from graphs to matrices. Next, we state that this $\varepsilon$-regularity is sufficient for obtaining a matrix analogue of the counting lemma for $2$-dimensional matrices but not for higher-dimensional cases. Finally, we introduce $\varepsilon$-regular patterns that allow us to deduce a multidimensional counting lemma.