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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
Published electronically: January 10, 2007
MathSciNet review: 2299792
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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

DOI: http://dx.doi.org/10.1090/S0025-5718-07-01961-8
PII: S 0025-5718(07)01961-8
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