The sumset phenomenon
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- by Renling Jin PDF
- Proc. Amer. Math. Soc. 130 (2002), 855-861 Request permission
Abstract:
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 $A$ and $B$ are large in terms of “measure”, then the sum $A+B$ 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 $A$ and $B$ of non–negative integers have positive upper or upper Banach density, then $A+B$ is piecewise syndetic.References
- Vitaly Bergelson, Ergodic Ramsey theory—an update, Ergodic theory of $\textbf {Z}^d$ actions (Warwick, 1993–1994) London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 1–61. MR 1411215, DOI 10.1017/CBO9780511662812.002
- H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
- 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
- Jin, Renling, Nonstandard Methods for Upper Banach Density Problems, to appear, The Journal of Number Theory. http://math.cofc.edu/faculty/jin/research/publication.html
- Jin, Renling, Standardizing Nonstandard Methods for Upper Banach Density Problems, to appear, DIMACS Series, Unusual Applications of Number Theory. http://math.cofc.edu/faculty/jin/research/publication.html
- 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
- Lindstrom, T., An invitation to nonstandard analysis, in Nonstandard Analysis and Its Application, ed. by N. Cutland, Cambridge University Press, 1988.
- Melvyn B. Nathanson, Additive number theory, Graduate Texts in Mathematics, vol. 164, Springer-Verlag, New York, 1996. The classical bases. MR 1395371, DOI 10.1007/978-1-4757-3845-2
Additional Information
- Renling Jin
- Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424
- Email: jinr@cofc.edu
- 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.
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 855-861
- MSC (2000): Primary 03H05, 03H15; Secondary 11B05, 11B13, 28E05
- DOI: https://doi.org/10.1090/S0002-9939-01-06088-9
- MathSciNet review: 1866042