Abstract
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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This paper is dedicated to Dona Strauss on the occasion of her 75th birthday
The author acknowledges financial support from the Austrian Science Fund (FWF) under grant P21209.
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Beiglböck, M. An ultrafilter approach to Jin’s theorem. Isr. J. Math. 185, 369–374 (2011). https://doi.org/10.1007/s11856-011-0114-5
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DOI: https://doi.org/10.1007/s11856-011-0114-5