Arithmetic Progressions in Abundance by Combinatorial Tools

TL;DR

This paper provides a combinatorial proof that any piecewise syndetic set contains arbitrarily long arithmetic progressions, connecting algebraic and combinatorial approaches in additive number theory.

Contribution

It introduces a purely combinatorial argument deriving the existence of arithmetic progressions from van der Waerden's Theorem, simplifying previous algebraic proofs.

Findings

Piecewise syndetic sets contain k-term arithmetic progressions for any k

A combinatorial proof can replace algebraic methods in this context

The approach simplifies understanding of structure within large sets

Abstract

Using the algebraic structure of the Stone-Cech compactification of the integers, Furstenberg and Glasner proved that for arbitrary k, every piecewise syndetic set contains a piecewise syndetic set of k-term arithmetic progressions. We present a purely combinatorial argument which allows to derive this result directly from van der Waerden's Theorem.