An Elementary Proof of Jin's Theorem with a Bound

TL;DR

This paper offers a straightforward, elementary proof of Jin's theorem, providing an explicit bound on the number of shifts needed to cover a thick set, avoiding advanced mathematical tools.

Contribution

It introduces a simple, elementary proof of Jin's theorem with an explicit bound, eliminating the need for nonstandard analysis or ergodic theory.

Findings

Provides an explicit bound of 1/c for covering shifts

Eliminates reliance on advanced mathematical tools

Simplifies the proof of Jin's theorem

Abstract

We present a short proof of Jin's theorem 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 given proof provides the explicit bound 1/c where c=BD(A)*BD(B) to the number of shifts of A+B that are needed to cover a thick set.