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Lower bounds for Z-numbers


Authors: Arturas Dubickas and Michael J. Mossinghoff
Journal: Math. Comp. 78 (2009), 1837-1851
MSC (2000): Primary 11K31; Secondary 11J71, 11Y35
Published electronically: January 23, 2009
MathSciNet review: 2501079
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ p/q$ be a rational noninteger number with $ p>q\geq2$. A real number $ \lambda>0$ is a $ Z_{p/q}$-number if $ \{\lambda(p/q)^n\}<1/q$ for every nonnegative integer $ n$, where $ \{x\}$ denotes the fractional part of $ x$. We develop several algorithms to search for $ Z_{p/q}$-numbers, and use them to determine lower bounds on such numbers for several $ p$ and $ q$. It is shown, for instance, that if there is a $ Z_{3/2}$-number, then it is greater than $ 2^{57}$. We also explore some connections between these problems and some questions regarding iterated maps on integers.


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Additional Information

Arturas Dubickas
Affiliation: Department of Mathematics and Informatics, Vilnius University Naugarduko 24, LT-03225 Vilnius, Lithuania
Email: arturas.dubickas@mif.vu.lt

Michael J. Mossinghoff
Affiliation: Department of Mathematics, Davidson College, Davidson, North Carolina 28035-6996
Email: mimossinghoff@davidson.edu

DOI: https://doi.org/10.1090/S0025-5718-09-02211-X
Keywords: $Z$-numbers, distribution mod 1.
Received by editor(s): January 22, 2008
Received by editor(s) in revised form: August 7, 2008
Published electronically: January 23, 2009
Additional Notes: The research of the first author was partially supported by the Lithuanian State Science and Studies Foundation.
Article copyright: © Copyright 2009 American Mathematical Society