Arithmetic Progressions in Abundance by Combinatorial Tools
Mathias Beiglboeck

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.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Mathematical Dynamics and Fractals
