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Finite beta-expansions

Published online by Cambridge University Press:  19 September 2008

Christiane Frougny  and
Boris Solomyak
Christiane Frougny
Affiliation:
Université Paris 8 and Laboratoire Informatique Théorique et Programmation, Institut Blaise Pascal, 4 place Jussieu, 75252 Paris Cedex 05, France
Boris Solomyak
Affiliation:
Department of Mathematics GN-50, University of Washington, Seattle, Washington 98195, USA
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Abstract

We characterize numbers having finite β-expansions where β belongs to a certain class of Pisot numbers: when the β-expansion of 1 is equal to a1a2…am, where a1≥a2≥…≥am≥1 and when the β-expansion of 1 is equal to t1t2…tm(tm+1)ω where t1≥t2≥…≥tm>tm+1≥1.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 4 , December 1992 , pp. 713 - 723
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

[Be]Bertrand, A.. Développements en base de Pisot et répartition modulo 1. C.R. Acad. Sci., Paris 285 (1977), 419–421.Google Scholar
[B1]Blanchard, F.. β-expansions and symbolic dynamics. Theor. Comp. Sci. 65 (1989), 131–141.CrossRefGoogle Scholar
[B0]Boyd, D.. Salem numbers of degree four have periodic expansions. Number Theory, eds., de Coninck, J. H. and Levesque, C.. Walter de Gruyter, 1989, pp 57–64.Google Scholar
[Br]Brauer, A.. On algebraic equations with all but one root in the interior of the unit circle. Math. Nachr. 4 (1951), 250–257.CrossRefGoogle Scholar
[Fra]Fraenkel, A. S.. Systems of numeration. Amer. Math. Monthly 92(2) (1985), 105–114.CrossRefGoogle Scholar
[Fr1]Frougny, Ch.. Representations of numbers and finite automata. Math. Systems Theory 25 (1992). 37–60.CrossRefGoogle Scholar
[Fr2]Frougny, Ch.. How to write integers in non-integer base. LATIN 92, Saõ Paulo. Springer Lecture Notes in Computer Science 583 (1992), 154–164.CrossRefGoogle Scholar
[H]Handelman, D.. Spectral radii of primitive integral companion matrices. Contemp. Math. (1992).CrossRefGoogle Scholar
[P]Parry, W.. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungary 11 (1960), 401–416.CrossRefGoogle Scholar
[PT]Pethö, A. & Tichy, R.. On digit expansions with respect to linear recurrences. J. Number Theory 33 (1989), 243–256.CrossRefGoogle Scholar
[Pr]Praggastis, B.. University of Washington. PhD Thesis (1992).Google Scholar
[Q]Queffélec, M.. Substitution dynamical systems—spectral analysis. Springer Lecture Notes in Mathematics 1294 (1987).CrossRefGoogle Scholar
[R]Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungary 8 (1957), 477–493.CrossRefGoogle Scholar
[S]Schmidt, K.. On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980), 269–278.CrossRefGoogle Scholar
[Sa]Salem, R.. Algebraic Numbers and Fourier Analysis. D. C. Heath & Co.: Boston, 1963.Google Scholar
[Se]Seneta, E.. Non-negative Matrices. An Introduction to Theory and Applications. G. Allen & Unwin, London, 1973.Google Scholar
[So1]Solomyak, B.. Finite β-expansions and spectra of substitutions. Preprint, 1991.Google Scholar
[So2]Solomyak, B.. Substitutions, adic transformations, and beta-expansions. Contemp. Math. (1992).CrossRefGoogle Scholar
[T]Thurston, W.. Groups, tilings, and finite state automata. AMS Colloquium Lecture Notes, Boulder, 1989.Google Scholar