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Mathematics of Computation

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Lower bounds for Z-numbers

Authors: Artūras 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: Let $p/q$ be a rational noninteger number with $p>q\geq 2$. 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

Artūras Dubickas
Affiliation: Department of Mathematics and Informatics, Vilnius University Naugarduko 24, LT-03225 Vilnius, Lithuania

Michael J. Mossinghoff
Affiliation: Department of Mathematics, Davidson College, Davidson, North Carolina 28035-6996
MR Author ID: 630072
ORCID: 0000-0002-7983-5427

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