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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Optimal expansions in non-integer bases
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by Karma Dajani, Martijn de Vries, Vilmos Komornik and Paola Loreti PDF
Proc. Amer. Math. Soc. 140 (2012), 437-447 Request permission


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
  • Email:
  • Martijn de Vries
  • Affiliation: Tussen de Grachten 213, 1381 DZ Weesp, The Netherlands
  • Email:
  • Vilmos Komornik
  • Affiliation: Département de Mathématique, Université de Strasbourg, 7 rue René Descartes, 67084, Strasbourg Cedex, France
  • Email:
  • 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:
  • 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:
  • MathSciNet review: 2846313