Mathematics of Computation

Author(s): Jean-Paul Allouche; Christiane Frougny; Kevin G. Hare.
Journal: Math. Comp. 76 (2007), 1639-1660.
MSC (2000): Primary 11R06; Secondary 11A67
Posted: January 10, 2007
Retrieve article in: PDF

Abstract: We study Pisot numbers $\beta \in (1, 2)$ which are univoque, i.e., such that there exists only one representation of

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

. Furthermore we give the smallest limit point of the set of univoque Pisot numbers.

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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: 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
Posted: 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.
Copyright of article: Copyright 2007, American Mathematical Society

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