On univoque Pisot numbers
Authors:
Jean-Paul Allouche, Christiane Frougny and Kevin G. Hare
Journal:
Math. Comp. 76 (2007), 1639-1660
MSC (2000):
Primary 11R06; Secondary 11A67
DOI:
https://doi.org/10.1090/S0025-5718-07-01961-8
Published electronically:
January 10, 2007
MathSciNet review:
2299792
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Abstract | References | Similar Articles | Additional Information
Abstract: We study Pisot numbers $\beta \in (1, 2)$ which are univoque, i.e., such that there exists only one representation of $1$ as $1 = \sum _{n \geq 1} s_n\beta ^{-n}$, with $s_n \in \{0, 1\}$. We prove in particular that there exists a smallest univoque Pisot number, which has degree $14$. Furthermore we give the smallest limit point of the set of univoque Pisot numbers.
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Additional Information
Jean-Paul Allouche
Affiliation:
CNRS, LRI, Bâtiment 490, Université Paris-Sud, 91405 Orsay Cedex, France
Email:
allouche@lri.fr
Christiane Frougny
Affiliation:
LIAFA, CNRS UMR 7089, 2 place Jussieu, 75251 Paris Cedex 05, France, and Université Paris 8
Email:
Christiane.Frougny@liafa.jussieu.fr
Kevin G. Hare
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email:
kghare@math.uwaterloo.ca
Keywords:
Univoque,
Pisot number,
beta-expansion
Received by editor(s):
June 13, 2006
Received by editor(s) in revised form:
August 15, 2006
Published electronically:
January 10, 2007
Additional Notes:
Research of the first author was partially supported by MENESR, ACI NIM 154 Numération.
Research of the third author was supported, in part, by NSERC of Canada.
Article copyright:
© Copyright 2007
American Mathematical Society