Embeddability Properties of Difference Sets

TL;DR

This paper uses nonstandard analysis to establish embeddability properties of difference sets of integers, leading to improved bounds on their intersections and coverage of large intervals, advancing additive combinatorics.

Contribution

It introduces new embeddability results for difference sets using nonstandard analysis, refining previous theorems by Ruzsa and Jin with precise bounds.

Findings

Improved bounds on intersections of difference sets.

Enhanced understanding of how difference sets cover large intervals.

Application of nonstandard analysis to additive combinatorics.

Abstract

By using nonstandard analysis, we prove embeddability properties of difference sets $A-B$ of sets of integers. (A set $A$ is "embeddable" into $B$ if every finite configuration of $A$ has shifted copies in $B$.) As corollaries of our main theorem, we obtain improvements of results by I.Z. Ruzsa about intersections of difference sets, and of Jin's theorem (as refined by V. Bergelson, H. F\"urstenberg and B. Weiss), where a precise bound is given on the number of shifts of $A-B$ which are needed to cover arbitrarily large intervals.