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

DOI:
https://doi.org/10.1090/S0002-9947-02-03140-9

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:
https://doi.org/10.1090/S0002-9947-02-03140-9

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