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Generalized Hausdorff dimensions of sets of real numbers with zero entropy expansion

Published online by Cambridge University Press:  19 September 2011

CHRISTIAN MAUDUIT  and
CARLOS GUSTAVO MOREIRA
CHRISTIAN MAUDUIT
Affiliation:
Institut de Mathématiques de Luminy, 163, avenue de Luminy, 13288 Marseille Cedex 9, France (email: [email protected])
CARLOS GUSTAVO MOREIRA
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, RJ, Brasil
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Abstract

The complexity function of an infinite word w on a finite alphabet A is the sequence counting, for each non-negative integer n, the number of words of length n on the alphabet A that are factors of the infinite word w. Let f be a given function with subexponential growth. The goal of this work is to estimate the generalized Hausdorff dimensions of the set of real numbers whose q-adic expansion has a complexity function bounded by f and the set of real numbers whose continued fraction expansion is bounded by q and has a complexity function bounded by f.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 3 , June 2012 , pp. 1073 - 1089
Copyright
Copyright © Cambridge University Press 2011

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