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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
DOI: https://doi.org/10.1090/S0002-9947-02-03122-7
Published electronically: August 21, 2002
MathSciNet review: 1928077
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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 $0$.


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Additional Information

Renling Jin
Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424
Email: jinr@cofc.edu

DOI: https://doi.org/10.1090/S0002-9947-02-03122-7
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