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Arithmetic progressions in abundance by combinatorial tools
Author(s):
Mathias
Beiglböck
Journal:
Proc. Amer. Math. Soc.
137
(2009),
3981-3983.
MSC (2000):
Primary 05D10
Posted:
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  , every piecewise syndetic set contains a piecewise syndetic set of  -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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- [BeHi01]
- V. Bergelson and N. Hindman.
Partition regular structures contained in large sets are abundant.
J. Combin. Theory Ser. A, 93(1):18-36, 2001. MR 1807110 (2002i:05116) - [FuGl98]
- H. Furstenberg and E. Glasner.
Subset dynamics and van der Waerden's theorem.
In Topological dynamics and applications (Minneapolis, MN, 1995), volume 215 of Contemp. Math., pages 197-203. Amer. Math. Soc., Providence, RI, 1998. MR 1603189 (99d:11010) - [HiLS02]
- N. Hindman, I. Leader, and D. Strauss.
Image partition regular matrices--bounded solutions and preservation of largeness.
Discrete Math., 242(1-3):115-144, 2002. MR 1874760 (2002j:05146) - [Waer27]
- B. van der Waerden.
Beweis einer Baudetschen Vermutung.
Nieuw Arch. Wisk., 15:212-216, 1927.
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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:
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
Posted:
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 of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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