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Optimal expansions in non-integer bases


Authors: Karma Dajani, Martijn de Vries, Vilmos Komornik and Paola Loreti
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
Published electronically: July 1, 2011
MathSciNet review: 2846313
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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$.


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  • 1. K. Dajani, M. de Vries, Invariant densities for random $ \beta$-expansions, J. Eur. Math. Soc. 9 (2007), 157-176. MR 2283107 (2007j:37008)
  • 2. K. Dajani, C. Kalle, A natural extension for the greedy $ \beta$-transformation with three arbitrary digits, Acta Math. Hungar. 125 (2009), 21-45. MR 2564418 (2010j:37010)
  • 3. K. Dajani, C. Kraaikamp, From greedy to lazy expansions and their driving dynamics, Expo. Math. 20 (2002), 315-327. MR 1940010 (2003h:11089)
  • 4. Z. Daróczy, I. Kátai, Generalized number systems in the complex plane, Acta Math. Hungar. 51 (1988), 409-416. MR 956992 (90f:11017)
  • 5. M. de Vries, V. Komornik, Unique expansions of real numbers, Adv. Math. 221 (2009), 390-427. MR 2508926 (2011a:11010)
  • 6. M. de Vries, V. Komornik, A two-dimensional univoque set, Fund. Math. 212 (2011), 175-189.
  • 7. P. Erdős, M. Horváth, I. Joó, On the uniqueness of the expansions $ 1=\sum q^{-n_i}$, Acta Math. Hungar. 58 (1991), 333-342. MR 1153488 (93e:11012)
  • 8. P. Glendinning, N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001), 535-543. MR 1851269 (2002i:11009)
  • 9. P. Góra, Invariant densities for piecewise linear maps of the unit interval, Ergodic Theory Dynam. Systems 29 (2009), 1549-1583. MR 2545017 (2010i:37002)
  • 10. V. Komornik, P. Loreti, On the topological structure of univoque sets, J. Number Theory 122 (2007), 157-183. MR 2287118 (2007m:11009)
  • 11. V. Komornik, P. Loreti, Universal expansions in negative and complex bases, Integers 10 (2010), 669-679.
  • 12. C. Kopf, Invariant measures for piecewise linear transformations of the interval, Appl. Math. Comput. 39 (1990), 123-144. MR 1071209 (91i:58081)
  • 13. A. Lasota, J. A. Yorke, Exact dynamical systems and the Frobenius-Perron operator, Trans. Amer. Math. Soc. 273 (1982), 375-384. MR 664049 (84d:28023)
  • 14. W. Parry, On the $ \beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416. MR 0142719 (26:288)
  • 15. M. Pedicini, Greedy expansions and sets with deleted digits, Theoret. Comput. Sci. 332 (2005), 313-336. MR 2122508 (2005k:11013)
  • 16. A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493. MR 0097374 (20:3843)
  • 17. N. Sidorov, Almost every number has a continuum of $ \beta$-expansions, Amer. Math. Monthly 110 (2003), 838-842. MR 2024754 (2004i:11085)

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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

DOI: https://doi.org/10.1090/S0002-9939-2011-11226-7
Keywords: Greedy expansion, beta-expansion, ergodicity, invariant measure
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
Article copyright: © Copyright 2011 American Mathematical Society

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