On gaps in Rényi $\beta$-expansions of unity for $\beta >1$ an algebraic number

Verger-Gaugry, Jean-Louis

Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2565-2579.

Soit $\beta >1$ un nombre algébrique. Nous étudions les plages de zéros (“lacunes”) dans le $\beta$-développement de Rényi  $d_{\beta }\left(1\right)$ de l’unité qui contrôle l’ensemble ${\mathbb{Z}}_{\beta }$ des $\beta$-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 $d_{\beta }\left(1\right)$ présentent une “lacunarité” asymptotiquement bornée supérieurement par  $log\left(\mathrm{M}\right(\beta \left)\right)/log\left(\beta \right)$, où  $\mathrm{M}\left(\beta \right)$  est la mesure de Mahler de  $\beta$. La preuve de ce résultat fournit de manière naturelle une nouvelle classification des nombres algébriques $>1$ en classes appelées Q${}_i^{\left(j\right)}$ 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 $d_{\beta }\left(1\right)$. Comme corollaire, tous les nombres de Salem sont dans la classe C${}_1\cup$Q${}_0^{\left(1\right)}\cup$ Q${}_0^{\left(2\right)}\cup$ Q${}_0^{\left(3\right)}$ ; ce résultat est également obtenu par un théorème récent qui généralise le théorème de Thue-Siegel-Roth donné par Corvaja.

Let $\beta >1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $\beta$-expansion  $d_{\beta }\left(1\right)$ of unity which controls the set ${\mathbb{Z}}_{\beta }$ of $\beta$-integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in $d_{\beta }\left(1\right)$ are shown to exhibit a “gappiness” asymptotically bounded above by  $log\left(\mathrm{M}\right(\beta \left)\right)/log\left(\beta \right)$, where  $\mathrm{M}\left(\beta \right)$  is the Mahler measure of  $\beta$. The proof of this result provides in a natural way a new classification of algebraic numbers $>1$ with classes called Q${}_i^{\left(j\right)}$ 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 $d_{\beta }\left(1\right)$. As a corollary, all Salem numbers are in the class C${}_1\cup$Q${}_0^{\left(1\right)}\cup$ Q${}_0^{\left(2\right)}\cup$ Q${}_0^{\left(3\right)}$ ; this result is also directly proved using a recent generalization of the Thue-Siegel-Roth Theorem given by Corvaja.

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
Mot clés : Beta-entier, beta-numération, nombre de Pisot, nombre de Salem, nombre de Perron, mesure de Mahler, approximation Diophantienne, série de Mahler, quasicristal mathématique

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

[2] Allouche, J.-P.; Cosnard, M. The Komornik-Loreti constant is tanscendental, Amer. Math. Monthly, Volume 107 (2000), pp. 448-449 | DOI | MR | Zbl

[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 | Zbl

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

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

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

[7] Bertrand-Mathis, A. Développements en base $\theta$ et répartition modulo $1$ de la suite $\left(x{\theta }^n\right)$, Bull. Soc. Math. Fr., Volume 114 (1986), pp. 271-324 | Numdam | MR | Zbl

[8] Blanchard, F. $\beta$-expansions and Symbolic Dynamics, Theoret. Comput. Sci., Volume 65 (1989), pp. 131-141 | DOI | MR | Zbl

[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 | Zbl

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

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

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

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

[14] Chi, D. P.; Kwon, D. Sturmian words,  $\beta$-shifts, and transcendence, Theoret. Comput. Sci., Volume 321 (2004), pp. 395-404 | DOI | MR | Zbl

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

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

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

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

[19] Fabre, S. Substitutions et $\beta$-systèmes de numération, Theoret. Comput. Sci., Volume 137 (1995), pp. 219-236 | DOI | MR | Zbl

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

[38] Parry, W. On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hung., Volume 11 (1960), pp. 401-416 | DOI | MR | Zbl

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

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

[41] Schmeling, J. Symbolic dynamics for $\beta$-shift and self-normal numbers, Ergod. Th. & Dynam. Sys., Volume 17 (1997), pp. 675-694 | DOI | MR | Zbl

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

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

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

[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 (IRMA Lectures in Mathematics and Theoretical Physics), E.M.S. Publishing House, 2006 | Zbl

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