We present a short and self-contained proof of Jin's theorem about the piecewise syndeticity of dierence sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultralters, or other advanced tools. An explicit bound to the number of shifts that are needed to cover a thick set is provided. Precisely, we prove the following: If A and B are sets of integers having positive upper Banach densities a and b respectively, then there exists a finite set F of cardinality at most 1/ab such that (A-B) + F covers arbitrarily long intervals.
An elementary proof of Jin's theorem with a bound
DI NASSO, MAURO
2014-01-01
Abstract
We present a short and self-contained proof of Jin's theorem about the piecewise syndeticity of dierence sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultralters, or other advanced tools. An explicit bound to the number of shifts that are needed to cover a thick set is provided. Precisely, we prove the following: If A and B are sets of integers having positive upper Banach densities a and b respectively, then there exists a finite set F of cardinality at most 1/ab such that (A-B) + F covers arbitrarily long intervals.File in questo prodotto:
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