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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Near arithmetic progressions in sparse sets
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by Steven C. Leth PDF
Proc. Amer. Math. Soc. 134 (2006), 1579-1589 Request permission

Abstract:

Nonstandard methods are used to obtain results in combinatorial number theory. The main technique is to use the standard part map to translate density properties of subsets of $^{\ast }\mathbb {N}$ into Lebesgue measure properties on $[0,1]$. This allows us to obtain a simple condition on a standard sequence $A$ that guarantees the existence of intervals in arithmetic progression, all of which contain elements of $A$ with various uniform density conditions.
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Additional Information
  • Steven C. Leth
  • Affiliation: Department of Mathematical Sciences, University of Northern Colorado, Greeley, Colorado 80639
  • Email: steven.leth@unco.edu
  • Received by editor(s): August 31, 2004
  • Received by editor(s) in revised form: October 14, 2004, November 7, 2004, and January 5, 2005
  • Published electronically: December 2, 2005
  • Additional Notes: The author thanks the referee for identifying several errors in the original version
  • Communicated by: Carl G. Jockusch, Jr.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 1579-1589
  • MSC (2000): Primary 11B25, 11B05, 03H05, 03H15
  • DOI: https://doi.org/10.1090/S0002-9939-05-08141-4
  • MathSciNet review: 2204267