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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Abelian groups with layered tiles and the sumset phenomenon


Authors: Renling Jin and H. Jerome Keisler
Journal: Trans. Amer. Math. Soc. 355 (2003), 79-97
MSC (2000): Primary 20K99, 60B15, 22A05, 03H05; Secondary 11B05, 26E35, 28E05
Published electronically: September 6, 2002
MathSciNet review: 1928078
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove a generalization of the main theorem in Jin, The sumset phenomenon, about the sumset phenomenon in the setting of an abelian group with layered tiles of cell measures. Then we give some applications of the theorem for multi-dimensional cases of the sumset phenomenon. Several examples are given in order to show that the applications obtained are not vacuous and cannot be improved in various directions. We also give a new proof of Shnirel'man's theorem to illustrate a different approach (which uses the sumset phenomenon) to some combinatorial problems.


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Additional Information

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

H. Jerome Keisler
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: keisler@math.wisc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03140-9
PII: S 0002-9947(02)03140-9
Keywords: Abelian group, layered tiles of cell measures, the sumset phenomenon, upper Banach density.
Received by editor(s): October 10, 2001
Published electronically: September 6, 2002
Additional Notes: The first author’s research is supported in part by NSF grant DMS#0070407.
The second author’s research is supported in part by Vilas Trust Foundation.
Article copyright: © Copyright 2002 American Mathematical Society