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Proceedings of the American Mathematical Society

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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
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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Additional Information

Karma Dajani
Affiliation: Department of Mathematics, Utrecht University, 3508 TA Utrecht, The Netherlands

Martijn de Vries
Affiliation: Tussen de Grachten 213, 1381 DZ Weesp, The Netherlands

Vilmos Komornik
Affiliation: Département de Mathématique, Université de Strasbourg, 7 rue René Descartes, 67084, Strasbourg Cedex, France

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

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