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Abelian groups with layered tiles and the sumset phenomenon

Author(s): Renling Jin; H. Jerome Keisler
Journal: Trans. Amer. Math. Soc. 355 (2003), 79-97.
MSC (2000): Primary 20K99, 60B15, 22A05, 03H05; Secondary 11B05, 26E35, 28E05
Posted: September 6, 2002
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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.

References:

1.
Bergelson, V., Hindman, N., and McCutcheon, R., Notions of size and combinatorial properties of quotient sets in semigroups, Topology Proceedings, 23 (1998), pp. 23--60. MR 2001a:20114

2.
Chang, C. C. and Keisler, H. J., Model Theory, third edition, North-Holland, 1990. MR 91c:03026

3.
Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981. MR 82j:28010

4.
Jin, Renling, The Sumset phenomenon, Proc. Amer. Math. Soc., to appear.

5.
Keisler, H. Jerome and Leth, Steven C., Meager Sets on the Hyperfinite Time Line, The Journal of Symbolic Logic, 56 (1991), pp. 71-102. MR 93a:03074

6.
Henson, C. W., Foundations of nonstandard analysis: A gentle introduction to nonstandard extensions, in Nonstandard Analysis: Theory and Applications, ed. by N. J. Cutland, C. W. Henson, and L. Arkeryd, Kluwer Academic Publishers, Dordrecht, 1997. MR 99i:03085

7.
Lindstrøm, T., An invitation to nonstandard analysis, in Nonstandard Analysis and its Applications, ed. by N. Cutland, Cambridge University Press, Cambridge, 1988. CMP 21:05

8.
Nathanson, M. B., Additive Number Theory: The Classical Bases, Springer-Verlag, New York, 1996.

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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: 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
Posted: 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.
Copyright of article: Copyright 2002, American Mathematical Society

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