Embeddability Properties of Difference Sets

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.