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

 

Near arithmetic progressions in sparse sets


Author: Steven C. Leth
Journal: Proc. Amer. Math. Soc. 134 (2006), 1579-1589
MSC (2000): Primary 11B25, 11B05, 03H05, 03H15
Published electronically: December 2, 2005
MathSciNet review: 2204267
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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

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08141-4
PII: S 0002-9939(05)08141-4
Keywords: Nonstandard analysis, arithmetic progressions
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.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.