Arithmetic progressions in abundance by combinatorial tools

DOI10.1090/S0002-9939-09-09974-2zbMATH Open1175.05138arXiv0809.1709MaRDI QIDQ3648156

Mathias Beiglböck

Publication date: 24 November 2009

Published in: Proceedings of the American Mathematical Society (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/0809.1709

zbMATH Keywords

arithmetic progressionsvan der Waerden's theorempiecewise syndetic sets

Mathematics Subject Classification ID

Ramsey theory (05D10) Arithmetic progressions (11B25)

Cites Work

• Unnamed Item
• Partition regular structures contained in large sets are abundant
• Image partition regular matrices---bounded solutions and preservation of largeness

Related Items (7)

Unnamed Item ⋮ Some dynamical properties related to polynomials ⋮ Abundance of progressions in a commutative semigroup by elementary means ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Richness of arithmetic progression in commutative semigroup ⋮ Piecewise syndetic sets in \(\mathbb{N}^t\) and \(\mathbb{N}^X\)

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