On distribution of three-term arithmetic progressions in sparse subsets of F_p^n

TL;DR

This paper extends Szemeredi's regularity lemma to sparse subsets of F_p^n and applies it to analyze the distribution of three-term arithmetic progressions in such sets.

Contribution

It introduces a regularity lemma for sparse subsets of F_p^n and uses it to study the distribution of 3-term arithmetic progressions.

Findings

Regularity lemma adapted for sparse subsets of F_p^n

Distribution results for 3-term arithmetic progressions in sparse sets

Application of regularity lemma to probabilistic settings

Abstract

We prove a version of Szemeredi's regularity lemma for subsets of a typical random set in F_p^n. As an application, a result on the distribution of three-term arithmetic progressions in sparse sets is discussed.