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The sumset phenomenon
Author(s):
Renling
Jin
Journal:
Proc. Amer. Math. Soc.
130
(2002),
855-861.
MSC (2000):
Primary 03H05, 03H15;
Secondary 11B05, 11B13, 28E05
Posted:
June 8, 2001
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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  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.
References:
-
- 1.
- Bergelson, Vitaly, Ergodic Ramsey Theory-an Update, Ergodic theory of
 actions (Warwick, 1993-1994) 1-61, London Mathematical Society Lecture Note Ser. 228, Cambridge University Press, Cambridge, 1996. MR 98g:28017 - 2.
- Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981. MR 82j:28010
- 3.
- Henson, C. W., Foundations of nonstandard analysis-A gentle introduction to nonstandard extension, in Nonstandard Analysis: Theory and Applications, ed. by N. J. Cutland, C. W. Henson, and L. Arkeryd, Kluwer Academic Publishers, 1997. MR 99i:03085
- 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.
- Keisler, H. Jerome and Leth, Steven C., Meager Sets on the Hyperfinite Time Line, The Journal of Symbolic Logic, Vol. 56 (1991), pp. 71-102. MR 93a:03074
- 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.
- Nathanson, Melvyn B., Additive Number Theory-the Classical Bases, Springer, 1996. MR 97e:11004
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Additional Information:
Renling
Jin
Affiliation:
Department of Mathematics, College of Charleston, Charleston, South Carolina 29424
Email:
jinr@cofc.edu
DOI:
10.1090/S0002-9939-01-06088-9
PII:
S 0002-9939(01)06088-9
Received by editor(s):
September 14, 1999
Received by editor(s) in revised form:
August 9, 2000
Posted:
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 of article:
Copyright
2001,
American Mathematical Society
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