BETA CANTOR SERIES EXPANSION AND ADMISSIBLE SEQUENCES

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DOI:

https://doi.org/10.14311/AP.2020.60.0214

Keywords:

Beta expansion, Q-Cantor series expansion, numeration system, admissibility

Abstract

We introduce a numeration system, called the beta Cantor series expansion, 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 beta Cantor series expansion 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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Published

2020-07-01

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Articles