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

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 and are large in terms of ``measure'', then the sum 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 and of non-negative integers have positive upper or upper Banach density, then 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:
https://doi.org/10.1090/S0002-9939-01-06088-9

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