On univoque Pisot numbers

Allouche, Jean-Paul; Frougny, Christiane; Hare, Kevin

doi:10.1090/S0025-5718-07-01961-8

On univoque Pisot numbers
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by Jean-Paul Allouche, Christiane Frougny and Kevin G. Hare PDF

Math. Comp. 76 (2007), 1639-1660 Request permission

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.

References

J.-P. Allouche, Théorie des Nombres et Automates, Thèse d’État, Bordeaux, 1983.
Jean-Paul Allouche, Itérations de fonctions unimodales et suites engendrées par automates, Seminar on Harmonic Analysis, 1981–1982, Publ. Math. Orsay 83, vol. 2, Univ. Paris XI, Orsay, 1983, pp. 1–6 (French). MR 710675
Jean-Paul Allouche and Michel Cosnard, The Komornik-Loreti constant is transcendental, Amer. Math. Monthly 107 (2000), no. 5, 448–449. MR 1763399, DOI 10.2307/2695302
J.-P. Allouche and M. Cosnard, Non-integer bases, iteration of continuous real maps, and an arithmetic self-similar set, Acta Math. Hungar. 91 (2001), no. 4, 325–332. MR 1912007, DOI 10.1023/A:1010667918943
Jean-Paul Allouche and Jeffrey Shallit, The ubiquitous Prouhet-Thue-Morse sequence, Sequences and their applications (Singapore, 1998) Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 1999, pp. 1–16. MR 1843077
Mohamed Amara, Ensembles fermés de nombres algébriques, Ann. Sci. École Norm. Sup. (3) 83 (1966), 215–270 (1967) (French). MR 0237459
M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, and J.-P. Schreiber, Pisot and Salem numbers, Birkhäuser Verlag, Basel, 1992. With a preface by David W. Boyd. MR 1187044, DOI 10.1007/978-3-0348-8632-1
Anne Bertrand, Développements en base de Pisot et répartition modulo $1$, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 6, A419–A421 (French, with English summary). MR 447134
Peter Borwein, Computational excursions in analysis and number theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 10, Springer-Verlag, New York, 2002. MR 1912495, DOI 10.1007/978-0-387-21652-2
David W. Boyd, Pisot and Salem numbers in intervals of the real line, Math. Comp. 32 (1978), no. 144, 1244–1260. MR 491587, DOI 10.1090/S0025-5718-1978-0491587-8
David W. Boyd, Pisot numbers in the neighbourhood of a limit point. I, J. Number Theory 21 (1985), no. 1, 17–43. MR 804914, DOI 10.1016/0022-314X(85)90010-1
David W. Boyd, Pisot numbers in the neighbourhood of a limit point. I, J. Number Theory 21 (1985), no. 1, 17–43. MR 804914, DOI 10.1016/0022-314X(85)90010-1
David W. Boyd, Salem numbers of degree four have periodic expansions, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 57–64. MR 1024551
David W. Boyd, On beta expansions for Pisot numbers, Math. Comp. 65 (1996), no. 214, 841–860. MR 1325863, DOI 10.1090/S0025-5718-96-00693-X
David W. Boyd, On the beta expansion for Salem numbers of degree $6$, Math. Comp. 65 (1996), no. 214, 861–875, $S$29–$S$31. MR 1333306, DOI 10.1090/S0025-5718-96-00700-4
Karma Dajani and Cor Kraaikamp, From greedy to lazy expansions and their driving dynamics, Expo. Math. 20 (2002), no. 4, 315–327. MR 1940010, DOI 10.1016/S0723-0869(02)80010-X
Zoltán Daróczy and Imre Kátai, Univoque sequences, Publ. Math. Debrecen 42 (1993), no. 3-4, 397–407. MR 1229687
Zoltán Daróczy and Imre Kátai, On the structure of univoque numbers, Publ. Math. Debrecen 46 (1995), no. 3-4, 385–408. MR 1336377
J. Dufresnoy and Ch. Pisot, Etude de certaines fonctions méromorphes bornées sur le cercle unité. Application à un ensemble fermé d’entiers algébriques, Ann. Sci. Ecole Norm. Sup. (3) 72 (1955), 69–92 (French). MR 0072902
Pál Erdös, István Joó, and Vilmos Komornik, Characterization of the unique expansions $1=\sum ^\infty _{i=1}q^{-n_i}$ and related problems, Bull. Soc. Math. France 118 (1990), no. 3, 377–390 (English, with French summary). MR 1078082
Paul Glendinning and Nikita Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001), no. 4, 535–543. MR 1851269, DOI 10.4310/MRL.2001.v8.n4.a12
Vilmos Komornik and Paola Loreti, Unique developments in non-integer bases, Amer. Math. Monthly 105 (1998), no. 7, 636–639. MR 1633077, DOI 10.2307/2589246
Vilmos Komornik, Paola Loreti, and Attila Pethő, The smallest univoque number is not isolated, Publ. Math. Debrecen 62 (2003), no. 3-4, 429–435. Dedicated to Professor Lajos Tamássy on the occasion of his 80th birthday. MR 2008106
M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 90, Cambridge University Press, Cambridge, 2002. A collective work by Jean Berstel, Dominique Perrin, Patrice Seebold, Julien Cassaigne, Aldo De Luca, Steffano Varricchio, Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, Veronique Bruyere, Christiane Frougny, Filippo Mignosi, Antonio Restivo, Christophe Reutenauer, Dominique Foata, Guo-Niu Han, Jacques Desarmenien, Volker Diekert, Tero Harju, Juhani Karhumaki and Wojciech Plandowski; With a preface by Berstel and Perrin. MR 1905123, DOI 10.1017/CBO9781107326019
R. C. Lyndon and M. P. Schützenberger, The equation $a^{M}=b^{N}c^{P}$ in a free group, Michigan Math. J. 9 (1962), 289–298. MR 162838
W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416 (English, with Russian summary). MR 142719, DOI 10.1007/BF02020954
A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. MR 97374, DOI 10.1007/BF02020331
R. Salem, Power series with integral coefficients, Duke Math. J. 12 (1945), 153–172. MR 11720
Klaus Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), no. 4, 269–278. MR 576976, DOI 10.1112/blms/12.4.269
Faouzia Talmoudi, Sur les nombres de $S\cap [1,\,2]$, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 16, A969–A971 (French, with English summary). MR 507210
Faouzia Lazami Talmoudi, Sur les nombres de $S\cap [1,2[$, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 10, A739–A741 (French, with English summary). MR 516773