The sumset phenomenon

Author:
Renling Jin

Journal:
Proc. Amer. Math. Soc. **130** (2002), 855-861

MSC (2000):
Primary 03H05, 03H15; Secondary 11B05, 11B13, 28E05

Published electronically:
June 8, 2001

MathSciNet review:
1866042

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Abstract | References | Similar Articles | Additional Information

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 and are large in terms of ``measure'', then the sum 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 and of non-negative integers have positive upper or upper Banach density, then is piecewise syndetic.

**1.**Vitaly Bergelson,*Ergodic Ramsey theory—an update*, Ergodic theory of 𝑍^{𝑑} actions (Warwick, 1993–1994) London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 1–61. MR**1411215**, 10.1017/CBO9780511662812.002**2.**H. Furstenberg,*Recurrence in ergodic theory and combinatorial number theory*, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR**603625****3.**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****4.**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**5.**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**6.**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**, 10.2307/2274905**7.**Lindstrom, T.,*An invitation to nonstandard analysis*, in**Nonstandard Analysis and Its Application**, ed. by N. Cutland, Cambridge University Press, 1988. CMP**21:05****8.**Melvyn B. Nathanson,*Additive number theory*, Graduate Texts in Mathematics, vol. 164, Springer-Verlag, New York, 1996. The classical bases. MR**1395371**

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Additional Information

**Renling Jin**

Affiliation:
Department of Mathematics, College of Charleston, Charleston, South Carolina 29424

Email:
jinr@cofc.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-06088-9

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.

Article copyright:
© Copyright 2001
American Mathematical Society