Arithmetic progressions in abundance by combinatorial tools
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- by Mathias Beiglböck
- Proc. Amer. Math. Soc. 137 (2009), 3981-3983
- DOI: https://doi.org/10.1090/S0002-9939-09-09974-2
- Published electronically: July 17, 2009
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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.
References
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- H. Furstenberg and E. Glasner, Subset dynamics and van der Waerden’s theorem, Topological dynamics and applications (Minneapolis, MN, 1995) Contemp. Math., vol. 215, Amer. Math. Soc., Providence, RI, 1998, pp. 197–203. MR 1603189, DOI 10.1090/conm/215/02941
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- B. van der Waerden. Beweis einer Baudetschen Vermutung. Nieuw Arch. Wisk., 15:212–216, 1927.
Bibliographic Information
- Mathias Beiglböck
- Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria
- Email: mathias.beiglboeck@univie.ac.at
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 3981-3983
- MSC (2000): Primary 05D10
- DOI: https://doi.org/10.1090/S0002-9939-09-09974-2
- MathSciNet review: 2538557