## Lower bounds for Z-numbers

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- by Artūras Dubickas and Michael J. Mossinghoff PDF
- Math. Comp.
**78**(2009), 1837-1851 Request permission

## 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
- Email: arturas.dubickas@mif.vu.lt
**Michael J. Mossinghoff**- Affiliation: Department of Mathematics, Davidson College, Davidson, North Carolina 28035-6996
- MR Author ID: 630072
- ORCID: 0000-0002-7983-5427
- Email: mimossinghoff@davidson.edu
- 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.
- © Copyright 2009 American Mathematical Society
- Journal: Math. Comp.
**78**(2009), 1837-1851 - MSC (2000): Primary 11K31; Secondary 11J71, 11Y35
- DOI: https://doi.org/10.1090/S0025-5718-09-02211-X
- MathSciNet review: 2501079