An ultrafilter approach to Jin's Theorem

Mathias Beiglboeck

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arXiv · 0908.2872

An ultrafilter approach to Jin's Theorem

Mathias Beiglboeck

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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.

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Mathias Beiglboeck. 2009-08-20. An ultrafilter approach to Jin's Theorem. https://arxiv.org/abs/0908.2872

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