# The sumset phenomenon

```@inproceedings{Jin2001TheSP,
title={The sumset phenomenon},
author={Renling Jin},
year={2001}
}```
Answering a problem posed by Keisler and Leth, we prove a theorem in non-standard analysis to reveal a phenomenon about sumsets, which says that if two sets A and B are large in terms of measure, then the sum A+B is not small in terms of order-topology. The theorem has several corollaries about sumset phenomenon in the standard world; these are described in sections 2-4. One of these is a new result in additive number theory; it says that if two sets A and B of non-negative integers have…
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,
Abundant configurations in sumsets with one dense summand
We analyze sumsets A+B = {a+b : a in A, b in B} where A,B are sets of integers, A is infinite, and B has positive upper Banach density. For each k, we show that A+B contains at least the expected
Nonstandard analysis and the sumset phenomenon in arbitrary amenable groups
• Mathematics
• 2012
Beiglbock, Bergelson and Fish proved that if subsets A, B of a countable discrete amenable group G have positive Banach densitiesand � respectively, then the product set AB is piecewise syndetic,
Sumsets of dense sets and sparse sets
R. Jin showed that whenever A and B are sets of integers having positive upper Banach density, the sumset A+B:= «a+b: a ∈ A, b ∈ B» is piecewise syndetic. This result was strengthened by Bergelson,
An Elementary Proof of Jin's Theorem with a Bound
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,
Dichotomy and infinite combinatorics: the theorems of Steinhaus and Ostrowski
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2010
Abstract We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and
Piecewise-Bohr Sets of Integers and Combinatorial Number Theory
• Mathematics
• 2006
We use ergodic-theoretical tools to study various notions of “large” sets of integers which naturally arise in theory of almost periodic functions, combinatorial number theory, and dynamics. Call a

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