BETA CANTOR SERIES EXPANSION AND ADMISSIBLE SEQUENCES

• Main Authors: Jonathan Caalim, Shiela Demegillo
• Format: Article
• Language: English
• Published: CTU Central Library 2020-07-01
• Series: Acta Polytechnica
• Subjects: beta expansion, q-cantor series expansion, numeration system, admissibility
• Online Access: https://ojs.cvut.cz/ojs/index.php/ap/article/view/5897

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.