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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Email: mathias.beiglboeck@univie.ac.at

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09974-2
PII: S 0002-9939(09)09974-2
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.