An ultrafilter approach to Jin's Theorem

Abstract

It is well known and not difficult to prove that if $C$ of integers has positive upper Banach density, the set of differences $C-C$ is syndetic, i.e. the length of gaps is uniformly bounded. More surprisingly, Renling Jin showed that whenever $A$ and $B$ have positive upper Banach density, then $A-B$ is piecewise syndetic. Jin's result follows trivially from the first statement provided that $B$ has large intersection with a shifted copy $A-n$ of $A$. Of course this will not happen in general if we consider shifts by integers, but the idea can be put to work if we allow "shifts by ultrafilters". As a consequence we obtain Jin's Theorem.