Three applications of coverings to difference patterns

TL;DR

This paper explores a simple covering technique that has diverse applications in density theorems and pattern conjectures involving set differences, including extensions, reductions, and quasirandom cases.

Contribution

It introduces a unifying covering approach that simplifies and extends results related to difference patterns in sets.

Findings

Extended density theorems to distance 2 patterns

Reduced complex statements to relative and quasirandom cases

Provided new insights into pattern realizations in sets

Abstract

We show that a conceptually simple covering technique has surprisingly rich applications to density theorems and conjectures on patterns in sets involving set differences. These applications fall into three categories: (i) analogues of these statements to distance $2$ versions of the pattern, (ii) reduction of these statements to relative versions, and (iii) reductions of these statements to a quasirandom case with respect to some quantities that affect the number of realisations of the pattern.