Abstract
Real numbers can be represented in an arbitrary base β > 1 using the transformation T β : x → βx (mod 1) of the unit interval; any real number x ∈ [0, 1] is then expanded into d β(x) = (xi)i≥1 where xi = ⌊βT i-1 β(x)⌋
The closure of the set of the expansions of real numbers of [0, 1] is a subshift of a ∈ ℕ a < β ℕ, called the beta-shift. This dynamical system is characterized by the beta-expansion of 1; in particular, it is of finite type if and only if d β(1) is finite; β is then called a simple beta-number.
We first compute the beta-expansion of 1 for any cubic Pisot number. Next we show that cubic simple beta-numbers are Pisot numbers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. Akiyama. Cubic Pisot units with finite beta expansions. In F. Halter-Koch and R.F. Tichy, editors, Algebraic Number Theory and Diophantine Analysis, 11–26. de Gruyter, 2000.
A. Bertrand. Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris, 285:419–421, 1977.
A. Bertrand-Mathis. Développement en base θ, répartition modulo 1 de la suite (xθ n)n≥0, langages codés et θ-shift. Bull. Soc. Math. France, 114:271–323, 1986.
F. Blanchard. β-expansions and symbolic dynamics. Theor. Comput. Sci., 65:131–141, 1989.
D. W. Boyd. Salem numbers of degree four have periodic expansions. In Number theory, pages 57–64. de Gruyter, 1989.
D. W. Boyd. On beta expansions for Pisot numbers. Mathematics of Computation, 65(214):841–860, 1996.
D. W. Boyd. On the beta expansion for Salem numbers of degree 6. Mathematics of Computation, 65(214):861–875, 1996.
L. Flatto, J. Lagarias, and B. Poonen. The zeta function of the beta transformation. Ergodic Theory Dynamical Systems, 14:237–266, 1994.
C. Frougny. Numeration Systems, chapter 7, in M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press, to appear, available at http://www-igm.univ-mlv.fr/~berstel/Lothaire/.
C. Frougny and B. Solomyak. Finite β-expansions. Ergodic Theory Dynamical Systems, 12:713–723, 1992.
M. Hollander. Greedy numeration systems and regularity. Theory of Computing Systems, 31:111–133, 1998.
S. Ito and Y. Takahashi. Markov subshifts and realization of β-expansions. J. Math. Soc. Japan, 26:33–55, 1974.
W. Parry. On the beta expansions of real numbers. Acta Math. Acad. Sci. Hung., 11:401–416, 1960.
A. Rényi. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung., 8:477–493, 1957.
K. Schmidt. On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc., 12:269–278, 1980.
B. Solomyak. Conjugates of beta-numbers and the zero-free domain for a class of analytic functions. Proc. London Math. Soc., 68(3):477–498, 1994.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bassino, F. (2002). Beta-Expansions for Cubic Pisot Numbers. In: Rajsbaum, S. (eds) LATIN 2002: Theoretical Informatics. LATIN 2002. Lecture Notes in Computer Science, vol 2286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45995-2_17
Download citation
DOI: https://doi.org/10.1007/3-540-45995-2_17
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43400-9
Online ISBN: 978-3-540-45995-8
eBook Packages: Springer Book Archive