An ultrafilter approach to Jin’s theorem
It is well known and not difficult to prove that if C ⊆ ℤ 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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