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Arithmetic progressions in abundance by combinatorial tools

Author: Mathias Beiglböck
Journal: Proc. Amer. Math. Soc. 137 (2009), 3981-3983
MSC (2000): Primary 05D10
Published electronically: July 17, 2009
MathSciNet review: 2538557
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Abstract: Using the algebraic structure of the Stone-Čech compactification of the integers, Furstenberg and Glasner proved that for arbitrary $ k\in\mathbb{N}$, every piecewise syndetic set contains a piecewise syndetic set of $ k$-term arithmetic progressions.

We present a purely combinatorial argument which allows us to derive this result directly from van der Waerden's Theorem.

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Additional Information

Mathias Beiglböck
Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria

Keywords: Arithmetic progressions, piecewise syndetic sets, van der Waerden's Theorem
Received by editor(s): September 10, 2008
Received by editor(s) in revised form: March 16, 2009
Published electronically: July 17, 2009
Additional Notes: The author gratefully acknowledges financial support from the Austrian Science Fund (FWF) under grants S9612 and p21209.
Communicated by: Jim Haglund
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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