Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11A63, 37A05, 37L40

Retrieve articles in all journals with MSC (2010): 11A63, 37A05, 37L40


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: http://dx.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