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Periodic expansion of one by Salem numbers

Part of: Topological dynamics Probabilistic theory: distribution modulo $1$; metric theory of algorithms Low-dimensional dynamical systems

Published online by Cambridge University Press:  14 October 2022

SHIGEKI AKIYAMA Open the ORCID record for SHIGEKI AKIYAMA [Opens in a new window]  and
HACHEM HICHRI
SHIGEKI AKIYAMA*
Affiliation:
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan
HACHEM HICHRI
Affiliation:
Département de Mathématiques (UR18ES15), Faculté des sciences de Monastir, Université de Monastir, Monastir 5019, Tunisie (e-mail: hichemhichri@yahoo.fr)
*
e-mail: akiyama@math.tsukuba.ac.jp
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Abstract

We show that for a Salem number $\beta $ of degree d, there exists a positive constant $c(d)$ where $\beta ^m$ is a Parry number for integers m of natural density $\ge c(d)$. Further, we show $c(6)>1/2$ and discuss a relation to the discretized rotation in dimension $4$.

Keywords

beta expansionperiodicitySalem numberdiscretized rotation

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37B10: Symbolic dynamics 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 9 , September 2023 , pp. 2841 - 2862
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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