Optimal expansions in non-integer bases
HTML articles powered by AMS MathViewer
- by Karma Dajani, Martijn de Vries, Vilmos Komornik and Paola Loreti PDF
- Proc. Amer. Math. Soc. 140 (2012), 437-447 Request permission
Abstract:
For a given positive integer $m$, let $A=\{0,1,\ldots ,m\}$ and $q \in (m,m+1)$. A sequence $(c_i)=c_1c_2 \ldots$ consisting of elements in $A$ is called an expansion of $x$ if $\sum _{i=1}^{\infty } c_i q^{-i}=x$. It is known that almost every $x$ belonging to the interval $[0,m/(q-1)]$ has uncountably many expansions. In this paper we study the existence of expansions $(d_i)$ of $x$ satisfying the inequalities $\sum _{i=1}^n d_iq^{-i} \ge \sum _{i=1}^n c_i q^{-i}$ , $n=1,2,\ldots ,$ for each expansion $(c_i)$ of $x$.References
- Karma Dajani and Martijn de Vries, Invariant densities for random $\beta$-expansions, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 1, 157–176. MR 2283107, DOI 10.4171/JEMS/76
- K. Dajani and C. Kalle, A natural extension for the greedy $\beta$-transformation with three arbitrary digits, Acta Math. Hungar. 125 (2009), no. 1-2, 21–45. MR 2564418, DOI 10.1007/s10474-009-8212-0
- 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
- Z. Daróczy and I. Kátai, Generalized number systems in the complex plane, Acta Math. Hungar. 51 (1988), no. 3-4, 409–416. MR 956992, DOI 10.1007/BF01903347
- Martijn de Vries and Vilmos Komornik, Unique expansions of real numbers, Adv. Math. 221 (2009), no. 2, 390–427. MR 2508926, DOI 10.1016/j.aim.2008.12.008
- M. de Vries, V. Komornik, A two-dimensional univoque set, Fund. Math. 212 (2011), 175–189.
- P. Erdős, M. Horváth, and I. Joó, On the uniqueness of the expansions $1=\sum q^{-n_i}$, Acta Math. Hungar. 58 (1991), no. 3-4, 333–342. MR 1153488, DOI 10.1007/BF01903963
- 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
- PawełGóra, Invariant densities for piecewise linear maps of the unit interval, Ergodic Theory Dynam. Systems 29 (2009), no. 5, 1549–1583. MR 2545017, DOI 10.1017/S0143385708000801
- Vilmos Komornik and Paola Loreti, On the topological structure of univoque sets, J. Number Theory 122 (2007), no. 1, 157–183. MR 2287118, DOI 10.1016/j.jnt.2006.04.006
- V. Komornik, P. Loreti, Universal expansions in negative and complex bases, Integers 10 (2010), 669–679.
- Christoph Kopf, Invariant measures for piecewise linear transformations of the interval, Appl. Math. Comput. 39 (1990), no. 2, 123–144. MR 1071209, DOI 10.1016/0096-3003(90)90027-Z
- A. Lasota and James A. Yorke, Exact dynamical systems and the Frobenius-Perron operator, Trans. Amer. Math. Soc. 273 (1982), no. 1, 375–384. MR 664049, DOI 10.1090/S0002-9947-1982-0664049-X
- 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
- Marco Pedicini, Greedy expansions and sets with deleted digits, Theoret. Comput. Sci. 332 (2005), no. 1-3, 313–336. MR 2122508, DOI 10.1016/j.tcs.2004.11.002
- 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
- Nikita Sidorov, Almost every number has a continuum of $\beta$-expansions, Amer. Math. Monthly 110 (2003), no. 9, 838–842. MR 2024754, DOI 10.2307/3647804
Additional Information
- Karma Dajani
- Affiliation: Department of Mathematics, Utrecht University, 3508 TA Utrecht, The Netherlands
- Email: k.dajani1@uu.nl
- Martijn de Vries
- Affiliation: Tussen de Grachten 213, 1381 DZ Weesp, The Netherlands
- Email: martijndevries0@gmail.com
- Vilmos Komornik
- Affiliation: Département de Mathématique, Université de Strasbourg, 7 rue René Descartes, 67084, Strasbourg Cedex, France
- Email: vilmos.komornik@math.unistra.fr
- Paola Loreti
- Affiliation: Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sezione di Matematica, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy
- Email: loreti@dmmm.uniroma1.it
- Received by editor(s): November 25, 2010
- Published electronically: July 1, 2011
- Additional Notes: Part of this work was done during the visit of the third author to the Department of Mathematics of the Delft Technical University. He is grateful for this invitation and for the excellent working conditions.
- Communicated by: Bryna Kra
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 437-447
- MSC (2010): Primary 11A63; Secondary 37A05, 37L40
- DOI: https://doi.org/10.1090/S0002-9939-2011-11226-7
- MathSciNet review: 2846313