An ultrafilter approach to Jin's Theorem

TL;DR

This paper introduces an ultrafilter-based method to prove Jin's Theorem, showing that the difference set of two positive density sets is piecewise syndetic, extending classical combinatorial number theory results.

Contribution

It presents a novel ultrafilter approach to Jin's Theorem, providing a new proof technique for difference set properties in additive combinatorics.

Findings

Proves Jin's Theorem using ultrafilters.

Establishes that the difference set of two positive density sets is piecewise syndetic.

Introduces an ultrafilter approach as a new proof method.

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.