Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-09-09974-2
Published electronically: July 17, 2009
MathSciNet review: 2538557
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [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.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 05D10

Retrieve articles in all journals with MSC (2000): 05D10


Additional Information

Mathias Beiglböck
Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria
Email: mathias.beiglboeck@univie.ac.at

DOI: https://doi.org/10.1090/S0002-9939-09-09974-2
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.

American Mathematical Society