Geodesic
An elementary proof of Jin's theorem with a bound
Mauro Di Nasso1
1University of Pisa, Italy
The electronic journal of combinatorics, Tome 21 (2014) no. 2
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We present a short and self-contained proof of Jin's theorem about the piecewise syndeticity of difference sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultrafilters, 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.
DOI : 10.37236/3846
Classification : 11B05, 11B13, 11B75
Mots-clés : sumsets, upper Banach density, piecewise syndetic set, Jin's theorem

Mauro Di Nasso  1

1 University of Pisa, Italy 
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Mauro Di Nasso. An elementary proof of Jin's theorem with a bound. The electronic journal of combinatorics, Tome 21 (2014) no. 2. doi: 10.37236/3846

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