A transcendence criterion for Cantor series

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the interval$[0,1)$, viewed as$\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilising our theorem. In particular, we give a Diophantine approximation interpretation of our scheme.

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Cantor series constructions contrasting two notions of normality

Monatshefte für Mathematik ◽

10.1007/s00605-010-0213-0 ◽

2010 ◽

Vol 164 (1) ◽

pp. 1-22 ◽

Cited By ~ 4

Author(s):

C. Altomare ◽

B. Mance

Keyword(s):

Cantor Series

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Multi-splitting and self-similarity of band gap structures in quasi-periodic plates of Cantor series

Applied Physics Letters ◽

10.1063/1.3687648 ◽

2012 ◽

Vol 100 (8) ◽

pp. 083501 ◽

Cited By ~ 13

Author(s):

Hong-Xing Ding ◽

Zhong-Hua Shen ◽

Xiao-Wu Ni ◽

Xue-Feng Zhu

Keyword(s):

Band Gap ◽

Self Similarity ◽

Cantor Series

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BETA CANTOR SERIES EXPANSION AND ADMISSIBLE SEQUENCES

Acta Polytechnica ◽

10.14311/ap.2020.60.0214 ◽

2020 ◽

Vol 60 (3) ◽

pp. 214-224

Author(s):

Jonathan Caalim ◽

Shiela Demegillo

Keyword(s):

Real Number ◽

Series Expansion ◽

Numeration System ◽

Integer Sequence ◽

Cantor Series ◽

Admissible Sequences

We introduce a numeration system, called the <em>beta Cantor series expansion</em>, that generalizes the classical positive and negative beta expansions by allowing non-integer bases in the Q-Cantor series expansion. In particular, we show that for a fix $\gamma \in \mathbb{R}$ and a sequence $B=\{\beta_i\}$ of real number bases, every element of the interval $x \in [\gamma,\gamma+1)$ has a <em>beta Cantor series expansion</em> with respect to B where the digits are integers in some alphabet $\mathcal{A}(B)$. We give a criterion in determining whether an integer sequence is admissible when $B$ satisfies some condition. We provide a description of the reference strings, namely the expansion of $\gamma$ and $\gamma+1$, used in the admissibility criterion.

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