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
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
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
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
...
...

References

SHOWING 1-10 OF 24 REFERENCES
Meager sets on the hyperfinite time line
In this paper we study notions of a “meager subset” of a hyper-finite set. We work within an ω-saturated nonstandard universe and fix a hyperfinite natural number Є *N∖N. We shall consider subsets of
Standardizing nonstandard methods for upper Banach density problems
  • Renling Jin
  • Mathematics
    Unusual Applications of Number Theory
  • 2000
TLDR
Many results in [J1, J2] about the addition of sets of positive upper Banach density are proven here using standard methods that are translated from the nonstandard methods used in this work.
Additive Number Theory The Classical Bases
The purpose of this book is to describe the classical problems in additive number theory, and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial
Nonstandard Methods for Upper Banach Density Problems
Abstract A general method is developed by using nonstandard analysis for formulating and proving a theorem about upper Banach density parallel to each theorem about Shnirel'man density or lower
Meager Sets on the Hyper nite Time Line
  • The Journal of Symbolic Logic, Vol
  • 1991
Ergodic Ramsey Theory–an Update, Ergodic theory of Zd actions (Warwick
  • London Mathematical Society Lecture Note Ser
  • 1996
On additive number theory
Foundations of nonstandard analysis { A gentle introduction tononstandard extension , in Nonstandard Analysis : Theory and Applications
  • Kluwer AcademicPublishers
  • 1997
...
...