Israel Journal of Mathematics

, 185:369 | Cite as

An ultrafilter approach to Jin’s theorem

Article

Abstract

It is well known and not difficult to prove that if C ⊆ ℤ has positive upper Banach density, the set of differences CC 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 AB is piecewise syndetic.

Jin’s result follows trivially from the first statement provided that B has large intersection with a shifted copy An 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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Copyright information

© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversity of ViennaWienAustria

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