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

 

The sumset phenomenon


Author: Renling Jin
Journal: Proc. Amer. Math. Soc. 130 (2002), 855-861
MSC (2000): Primary 03H05, 03H15; Secondary 11B05, 11B13, 28E05
Published electronically: June 8, 2001
MathSciNet review: 1866042
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Abstract | References | Similar Articles | Additional Information

Abstract:

Answering a problem posed by Keisler and Leth, we prove a theorem in non-standard analysis to reveal a phenomenon about sumsets, which says that if two sets $A$ and $B$ are large in terms of ``measure'', then the sum $A+B$ is not small in terms of ``order-topology''. The theorem has several corollaries about sumset phenomenon in the standard world; these are described in sections 2-4. One of these is a new result in additive number theory; it says that if two sets $A$ and $B$ of non-negative integers have positive upper or upper Banach density, then $A+B$ is piecewise syndetic.


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Additional Information

Renling Jin
Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424
Email: jinr@cofc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06088-9
PII: S 0002-9939(01)06088-9
Received by editor(s): September 14, 1999
Received by editor(s) in revised form: August 9, 2000
Published electronically: June 8, 2001
Additional Notes: This research was supported in part by a Ralph E. Powe Junior Faculty Enhancement Award from Oak Ridge Association Universities, a Faculty Research and Development Summer Grant from College of Charleston, and NSF grant DMS–#0070407.
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2001 American Mathematical Society