An Elementary Proof of Jin's Theorem with a Bound

Mauro Di Nasso

doi:10.37236/3846

Abstract

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.

Highlights

A beautiful result in additive and combinatorial number theory was proved in 2001 by R
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
The underlying ideas of the main constructions used in [5] are suitably translated into “standard” terms; the goal is to provide a short and elementary proof of Jin’s theorem, completely from scratch. (Here, by “elementary” we mean that no use is made the electronic journal of combinatorics 21(2) (2014), #P2.37 of nonstandard analysis, measure theory and ergodic theory, ultrafilters, or any other advanced tool; and by “from scratch” we mean that the arguments used are self-contained, with no references to other existing results in the literature.)

Summary

Introduction

A beautiful result in additive and combinatorial number theory was proved in 2001 by R. Several papers recently appeared where Jin’s theorem was re-proved and extended by using more familiar non-elementary tools; most notably, see the proofs in [4, 1] by means of ergodic theory, and the ultrafilter proof in [2]. In [5] I gave a different nonstandard proof of the result, where an explicit bound to the number of shifts of A + B that are needed to cover arbitrarily large intervals is given. (Here, by “elementary” we mean that no use is made the electronic journal of combinatorics 21(2) (2014), #P2.37 of nonstandard analysis, measure theory and ergodic theory, ultrafilters, or any other advanced tool; and by “from scratch” we mean that the arguments used are self-contained, with no references to other existing results in the literature.). Notation: By N we denote the set of positive integers. We shall write A − z (or A + z) to indicate the left-ward shift A − {z} (or the right-ward shift A + {z}, respectively)

Jin’s theorem with a bound

The elementary proof of Jin’s theorem

Final remarks and open problems