On gaps in Rényi β-expansions of unity for β>1 an algebraic number
Annales de l'Institut Fourier, Volume 56 (2006) no. 7, p. 2565-2579
Let β>1 be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi β-expansion  d β (1) of unity which controls the set β of β-integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in d β (1) are shown to exhibit a “gappiness” asymptotically bounded above by  log(M(β))/log(β), where  M(β)  is the Mahler measure of  β. The proof of this result provides in a natural way a new classification of algebraic numbers >1 with classes called Q i (j) which we compare to Bertrand-Mathis’s classification with classes C 1 to C 5 (reported in an article by Blanchard). This new classification relies on the maximal asymptotic “quotient of the gap” value of the “gappy” power series associated with d β (1). As a corollary, all Salem numbers are in the class C 1 Q 0 (1) Q 0 (2) Q 0 (3) ; this result is also directly proved using a recent generalization of the Thue-Siegel-Roth Theorem given by Corvaja.
Soit β>1 un nombre algébrique. Nous étudions les plages de zéros (“lacunes”) dans le β-développement de Rényi  d β (1) de l’unité qui contrôle l’ensemble β des β-entiers. En utilisant une version de l’inégalité de Liouville qui étend des théorèmes d’approximation de Mahler et de Güting, on montre que les plages de zéros dans d β (1) présentent une “lacunarité” asymptotiquement bornée supérieurement par  log(M(β))/log(β), où  M(β)  est la mesure de Mahler de  β. La preuve de ce résultat fournit de manière naturelle une nouvelle classification des nombres algébriques >1 en classes appelées Q i (j) que nous comparons à la classification de Bertrand-Mathis avec les classes C 1 à C 5 (reportée dans un article de Blanchard). Cette nouvelle classification repose sur la valeur asymptotique maximale du “quotient de lacune” de la série “lacunaire” associée à d β (1). Comme corollaire, tous les nombres de Salem sont dans la classe C 1 Q 0 (1) Q 0 (2) Q 0 (3)  ; ce résultat est également obtenu par un théorème récent qui généralise le théorème de Thue-Siegel-Roth donné par Corvaja.
DOI : https://doi.org/10.5802/aif.2250
Classification:  11B05,  11Jxx,  11J68,  11R06,  52C23
Keywords: Beta-integer, beta-numeration, PV number, Salem number, Perron number, Mahler measure, Diophantine approximation, Mahler’s series, mathematical quasicrystal
@article{AIF_2006__56_7_2565_0,
     author = {Verger-Gaugry, Jean-Louis},
     title = {On gaps in R\'enyi $\beta $-expansions of unity for $\beta > 1$ an algebraic number},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {7},
     year = {2006},
     pages = {2565-2579},
     doi = {10.5802/aif.2250},
     zbl = {pre05176579},
     mrnumber = {2290791},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2006__56_7_2565_0}
}
On gaps in Rényi $\beta $-expansions of unity for $\beta > 1$ an algebraic number. Annales de l'Institut Fourier, Volume 56 (2006) no. 7, pp. 2565-2579. doi : 10.5802/aif.2250. https://aif.centre-mersenne.org/item/AIF_2006__56_7_2565_0/

[1] Adamczewski, B. Transcendance “à la Liouville” de certains nombres réels, C. R. Acad. Sci. Paris, Tome 338 (2004) no. I, pp. 511-514 | MR 2057021 | Zbl 1046.11051

[2] Allouche, J.-P.; Cosnard, M. The Komornik-Loreti constant is tanscendental, Amer. Math. Monthly, Tome 107 (2000), p. 448-449 | Article | MR 1763399 | Zbl 0997.11052

[3] Allouche, J.-P.; Shallit, J.; Ding, C.; Helleseth, T.; Niederreiter, H. The ubiquitous Prouhet-Thue-Morse sequence, Sequences and Their Applications, Springer-Verlag, Proceedings of SETA’98 (1999), pp. 1-16 | MR 1843077 | Zbl 1005.11005

[4] Bertin, M.-J.; Decomps-Guilloux, A.; Grandet-Hugot, M.; Pathiaux-Delefosse, M.; Schreiber, J.-P. Pisot and Salem Numbers, Birkhaüser (1992) | MR 1187044 | Zbl 0772.11041

[5] Bertrand-Mathis, A. Questions diverses relatives aux systèmes codés : applications au   θ -shift (preprint)

[6] Bertrand-Mathis, A. Développements en base Pisot et répartition modulo  1, C. R. Acad. Sci. Paris, Tome 285 (1977) no. A, pp. 419-421 | MR 447134 | Zbl 0362.10040

[7] Bertrand-Mathis, A. Développements en base θ et répartition modulo 1 de la suite (xθ n ), Bull. Soc. Math. Fr., Tome 114 (1986), pp. 271-324 | Numdam | MR 878240 | Zbl 0628.58024

[8] Blanchard, F. β-expansions and Symbolic Dynamics, Theoret. Comput. Sci., Tome 65 (1989), pp. 131-141 | Article | MR 1020481 | Zbl 0682.68081

[9] Boyd, D. Salem numbers of degree four have periodic expansions, Théorie des Nombres - Number Theory, Walter de Gruyter & Co., Eds. J.M. de Koninck and C. Levesque, Berlin and New York (1989), pp. 57-64 | MR 1024551 | Zbl 0685.12004

[10] Boyd, D. On beta expansions for Pisot numbers, Math. Comp., Tome 65 (1996), pp. 841-860 | Article | MR 1325863 | Zbl 0855.11039

[11] Boyd, D. On the beta expansion for Salem numbers of degree 6, Math. Comp., Tome 65 (1996), pp. 861-875 | Article | MR 1333306 | Zbl 0848.11048

[12] Boyd, D. The beta expansions for Salem numbers, Organic Mathematics, A.M.S., Providence, RI (Canad. Math. Soc. Conf. Proc. 20) (1997), pp. 117-131 | MR 1483916 | Zbl 1053.11536

[13] Burdik, C.; Frougny, Ch.; Gazeau, J.-P.; Krejcar, R. Beta-integers as natural counting systems for quasicrystals, J. Phys. A: Math. Gen., Tome 31 (1998), pp. 6449-6472 | Article | MR 1644115 | Zbl 0941.52019

[14] Chi, D. P.; Kwon, D. Sturmian words,  β-shifts, and transcendence, Theoret. Comput. Sci., Tome 321 (2004), pp. 395-404 | Article | MR 2076154 | Zbl 1068.68112

[15] Corvaja, P. Autour du Théorème de Roth, Monath. Math., Tome 124 (1997), pp. 147-175 | Article | MR 1462860 | Zbl 0883.11033

[16] Corvaja, P.; Zannier, U. Some New Applications of the Subspace Theorem, Compositio Mathematica, Tome 131 (2002), pp. 319-340 | Article | MR 1905026 | Zbl 1010.11038

[17] Denker, M.; Grillenberger, C.; Sigmund, K. Ergodic Theory on compact spaces, Springer Lecture Notes in Math. 527 (1976) | MR 457675 | Zbl 0328.28008

[18] Elkharrat, A.; Frougny, C.; Gazeau, J.-P.; Verger-Gaugry, J.-L. Symmetry groups for beta-lattices, Theor. Comp. Sci., Tome 319 (2004), pp. 281-305 | Article | MR 2074957 | Zbl 1068.52028

[19] Fabre, S. Substitutions et β-systèmes de numération, Theoret. Comput. Sci., Tome 137 (1995), pp. 219-236 | Article | MR 1311222 | Zbl 0872.11017

[20] Flatto, L.; Lagarias, J.C.; Poonen, B. The zeta function of the beta transformation, Ergod. Th. and Dynam. Sys., Tome 14 (1994), pp. 237-266 | Article | MR 1279470 | Zbl 0843.58106

[21] Frougny, C.; Gazeau, J.-P.; Krejcar, R. Additive and multiplicative properties of point sets based on beta-integers, Theoret. Comput. Sci., Tome 303 (2003), pp. 491-516 | Article | MR 1990778 | Zbl 1036.11034

[22] Frougny, C.; Solomyak, B. Finite beta-expansions, Ergod. Theor. Dynam. Sys., Tome 12 (1992), pp. 713-723 | MR 1200339 | Zbl 0814.68065

[23] Frougny, Ch. Number Representation and Finite Automata, London Math. Soc. Lecture Note Ser., Tome 279 (2000), pp. 207-228 | MR 1776760 | Zbl 0976.11003

[24] Frougny, Ch. Numeration systems, 7, Algebraic Combinatorics on Words, Cambridge University Press (2003) | MR 1905123

[25] Gazeau, J.-P. Pisot-Cyclotomic Integers for Quasilattices, The Mathematics of Long-Range Aperiodic Order, Ed. R. V. Moody, Kluwer Academic Publisher, Dordrecht (1997), pp. 175-198 | MR 1460024 | Zbl 0887.11043

[26] Gazeau, J.-P.; Verger-Gaugry, J.-L. Geometric study of the set of β-integers for a Perron number and mathematical quasicrystals, J. Th. Nombres Bordeaux, Tome 16 (2004), pp. 1-25 | Numdam | MR 2145576 | Zbl 1075.11007

[27] Gazeau, J.-P.; Verger-Gaugry, J.-L. Diffraction spectra of weighted Delone sets on β -lattices with β a quadratic unitary Pisot number (2006) (Ann. Inst. Fourier) | Numdam

[28] Güting, R. Approximation of algebraic numbers by algebraic numbers, Michigan Math. J., Tome 8 (1961), pp. 149-159 | Article | MR 132722 | Zbl 0107.04203

[29] Komornik, V.; Loreti, P. Unique developments in non-integer bases, Amer. Math. Monthly, Tome 105 (1998), pp. 636-639 | Article | MR 1633077 | Zbl 0918.11006

[30] Lang, S. Fundamentals of Diophantine Geometry, Springer-Verlag, New York (1983), pp. 158-187 | MR 715605 | Zbl 0528.14013

[31] Leveque, W. J. Topics in Number Theory, Addison-Wesley, Tome II (1956), pp. 121-160 | MR 80682 | Zbl 0070.03804

[32] Lind, D. The entropies of topological Markov shifts and a related class of algebraic integers, Erg. Th. Dyn. Syst., Tome 4 (1984), pp. 283-300 | Article | MR 766106 | Zbl 0546.58035

[33] Lind, D. Matrices of Perron numbers, J. Number Theory, Tome 40 (1992), pp. 211-217 | Article | MR 1149738 | Zbl 0748.11051

[34] Lothaire, M. Algebraic Combinatorics on Words, Cambridge University Press (2003) | MR 1905123 | Zbl 1001.68093

[35] Mahler, K. Arithmetic properties of lacunary power series with integral coefficients, J. Austr. Math. Soc., Tome 5 (1965), pp. 56-64 | Article | MR 190094 | Zbl 0148.27703

[36] Nishioka, K. Algebraic independence by Mahler’s method and S-units equations, Compositio Math., Tome 92 (1994), pp. 87-110 | Numdam | MR 1275722 | Zbl 0802.11029

[37] Ostrowski, A. On representation of analytical functions by power series, J. London Math. Soc., Tome 1 (1926), pp. 251-263 ((Addendum), ibid 4 (1929), p. 32) | Article

[38] Parry, W. On the β-expansions of real numbers, Acta Math. Acad. Sci. Hung., Tome 11 (1960), pp. 401-416 | Article | MR 142719 | Zbl 0099.28103

[39] Pythéas Fogg, N. Substitutions in dynamics, arithmetics and combinatorics, Springer Lecture Notes in Math. 1794 (2003) | MR 1970385 | Zbl 1014.11015

[40] Rényi, A. Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung., Tome 8 (1957), pp. 477-493 | Article | MR 97374 | Zbl 0079.08901

[41] Schmeling, J. Symbolic dynamics for β-shift and self-normal numbers, Ergod. Th. & Dynam. Sys., Tome 17 (1997), pp. 675-694 | Article | MR 1452189 | Zbl 0908.58017

[42] Schmidt, K. On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc., Tome 12 (1980), pp. 269-278 | Article | MR 576976 | Zbl 0494.10040

[43] Schmidt, W. M. Diophantine Approximations and Diophantine Equations, Springer Lecture Notes in Math. 1467 (1991) | MR 1176315 | Zbl 0754.11020

[44] Solomyak, B. Conjugates of beta-numbers and the zero-free domain for a class of analytic functions, Proc. London Math. Soc. (3), Tome 68 (1993), pp. 477-498 | Article | MR 1262305 | Zbl 0820.30007

[45] Thurston, W. P. Groups, tilings, and finite state automata (Summer 1989) (A.M.S. Colloquium Lectures, Boulder)

[46] Verger-Gaugry, J.-L.; Nyssen, L. On self-similar finitely generated uniformly discrete (SFU-) sets and sphere packings, Number Theory and Physics, E.M.S. Publishing House (IRMA Lectures in Mathematics and Theoretical Physics) (2006) | Zbl 1170.52303

[47] Waldschmidt, M. Diophantine Approximation on Linear Algebraic Groups. Transcendence Properties of the Exponential Function in Several Variables, Springer-Verlag, Berlin (2000) | MR 1756786 | Zbl 0944.11024