Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T13:56:17.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Applications of Nonstandard Analysis in Additive Number Theory

Published online by Cambridge University Press:  15 January 2014

Renling Jin
Renling Jin*
Affiliation:
Department of Mathematics, College of Charleston, Charleston, Sc 29424, USAE-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 6 , Issue 3 , September 2000 , pp. 331 - 341
Copyright
Copyright © Association for Symbolic Logic 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Arkeryd, L., Cutland, N. J., and Henson, C. W., Nonstandard Analysis: Theory and Applications, Kluwer Academic Publishers, 1997, edited.Google Scholar
[2] Bergelson, Vitaly, Ergodic Ramsey theory–an update, Ergodic theory of zd actions (warwick, 1993–1994), London Mathematical Society Lecture Note Series 228, Cambridge Univ. Press, Cambridge, 1996, pp. 1–61.Google Scholar
[3] Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981.Google Scholar
[4] Halberstam, H. and Roth, K. F., Sequences, Oxford University Press, 1966.Google Scholar
[5] Henson, C. W. and Keisler, H.J., On the strength of nonstandard analysis, The Journal of Symbolic Logic, vol. 51 (1986), pp. 377–386.CrossRefGoogle Scholar
[6] Jin, Renling, Nonstandard methods for upper Banach density problems, in preparation, http://math.cofc.edu/faculty/jin/research/publication.html.Google Scholar
[7] Jin, Renling, Sumset phenomenon, in preparation, ,http://math.cofc.edu/faculty/jin/research/publication.html. Google Scholar
[8] 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.Google Scholar
[9] Leth, Steven C., Application of nonstandard models and Lebesgue measure to sequences of natural numbers, Transactions of the American Mathematical Society, vol. 307 (1988), no. 2, pp. 457–468.Google Scholar
[10] Leth, Steven C., Sequences in countable models of the natural numbers, Studia Logica, vol. 47 (1988), no. 3, pp. 243–263.Google Scholar
[11] Leth, Steven C., Some nonstandard methods in combinatorial number theory, Studia Logica, vol. 47 (1988), pp. 265–278.CrossRefGoogle Scholar
[12] Lindstrøm, Tom, An invitation to nonstandard analysis, Nonstandard analysis and its applications (Cutland, N., editor), Cambridge University Press, Cambridge, 1988, pp. 1–105.Google Scholar
[13] Nathanson, Melvyn B., Additive Number Theory–Inverse Problems and the Geometry of Sumsets, Springer, 1996.CrossRefGoogle Scholar
[14] Nathanson, Melvyn B., Additive Number Theory–the Classical Bases, Springer, 1996.Google Scholar
[15] Petersen, Karl, Ergodic Theory, Cambridge University Press, 1983.Google Scholar
[16] Ross, David A., Loeb measure and probability, Nonstandard Analysis: Theory and Applications (Cutland, N. J., Henson, C. W., and Arkeryd, L., editors), Kluwer Academic Publishers, 1997.Google Scholar