Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Inverse problem for upper asymptotic density


Author: Renling Jin
Journal: Trans. Amer. Math. Soc. 355 (2003), 57-78
MSC (2000): Primary 11B05, 11B13, 11U10, 03H15
Published electronically: August 21, 2002
MathSciNet review: 1928077
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a set $A$ of natural numbers, the structural properties are described when the upper asymptotic density of $2A+\{0,1\}$achieves the infimum of the upper asymptotic densities of all sets of the form $2B+\{0,1\}$, where the upper asymptotic density of $B$ is greater than or equal to the upper asymptotic density of $A$. As a corollary, we prove that if the upper asymptotic density of $A$ is less than $1$and the upper asymptotic density of $2A+\{0,1\}$ achieves the infimum, then the lower asymptotic density of $A$ must be $0$.


References [Enhancements On Off] (What's this?)

  • 1. Bilu, Y., Addition of sets of integers of positive density, Journal of Number Theory, 64 (1997), No. 2, 233--275. MR 98e:11013
  • 2. Halberstam, H. and Roth, K. F., Sequences, Oxford University Press, 1966. MR 35:1565
  • 3. Hegedüs, P., Piroska, G., and Ruzsa, I. Z., On the Schnirelmann density of sumsets, Publ. Math. Debrecen, 53/3-4 (1998), 333--345. MR 99k:11019
  • 4. Henson, C. W., Foundations of nonstandard analysis: A gentle introduction to nonstandard extensions, in Nonstandard Analysis: Theory and Applications, ed. by N. J. Cutland, C. W. Henson, and L. Arkeryd, Kluwer Academic Publishers, Dordrecht, 1997. MR 99i:03085
  • 5. Jin, Renling, Nonstandard methods for upper Banach density problems, Journal of Number Theory 91 (2001), No. 1, 20-38. MR 2002h:11011
  • 6. Jin, Renling, Standardizing nonstandard methods for upper Banach density problems, to appear, DIMACS series for the workshop on Unusual application of number theory, Jan. 2000.
  • 7. Lindstrøm, T., An invitation to nonstandard analysis, in Nonstandard Analysis and its Applications, ed. by N. Cutland, Cambridge University Press, 1988. CMP 21:05
  • 8. Nathanson, Melvyn B., Additive Number Theory-Inverse Problems and the Geometry of Sumsets, Springer, 1996.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11B05, 11B13, 11U10, 03H15

Retrieve articles in all journals with MSC (2000): 11B05, 11B13, 11U10, 03H15


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/S0002-9947-02-03122-7
PII: S 0002-9947(02)03122-7
Keywords: Upper asymptotic density, inverse problem, nonstandard analysis
Received by editor(s): July 1, 2001
Received by editor(s) in revised form: May 8, 2002
Published electronically: August 21, 2002
Additional Notes: The author was supported in part by the NSF grant DMS–#0070407
Article copyright: © Copyright 2002 American Mathematical Society