Abelian groups with layered tiles and the sumset phenomenon
HTML articles powered by AMS MathViewer
- by Renling Jin and H. Jerome Keisler PDF
- Trans. Amer. Math. Soc. 355 (2003), 79-97 Request permission
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
- Vitaly Bergelson, Neil Hindman, and Randall McCutcheon, Notions of size and combinatorial properties of quotient sets in semigroups, Proceedings of the 1998 Topology and Dynamics Conference (Fairfax, VA), 1998, pp. 23–60. MR 1743799
- C. C. Chang and H. J. Keisler, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990. MR 1059055
- H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
- Jin, Renling, The Sumset phenomenon, Proc. Amer. Math. Soc., to appear.
- H. Jerome Keisler and Steven C. Leth, Meager sets on the hyperfinite time line, J. Symbolic Logic 56 (1991), no. 1, 71–102. MR 1131731, DOI 10.2307/2274905
- C. Ward Henson, Foundations of nonstandard analysis: a gentle introduction to nonstandard extensions, Nonstandard analysis (Edinburgh, 1996) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 493, Kluwer Acad. Publ., Dordrecht, 1997, pp. 1–49. MR 1603228
- Lindstrøm, T., An invitation to nonstandard analysis, in Nonstandard Analysis and its Applications, ed. by N. Cutland, Cambridge University Press, Cambridge, 1988.
- Nathanson, M. B., Additive Number Theory: The Classical Bases, Springer–Verlag, New York, 1996.
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
- 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. - © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1928078