Unique double base expansions
Wolfgang Steiner (CNRS, Université de Paris)
Abstract: For pairs of real bases $\beta_0, \beta_1 > 1$, we study expansions of the form $\sum_{k=1}^\infty i_k / (\beta_{i_1} \beta_{i_2} \cdots \beta_{i_k})$ with digits $i_k \in \{0,1\}$. We characterise the pairs admitting non-trivial unique expansions as well as those admitting uncountably many unique expansions, extending recent results of Neunhäuserer (2021) and Zou, Komornik and Lu (2021). Similarly to the study of unique $\beta$-expansions with three digits by the speaker (2020), this boils down to determining the cardinality of binary shifts defined by lexicographic inequalities. Labarca and Moreira (2006) characterised when such a shift is empty, at most countable or uncountable, depending on the position of the lower and upper bounds with respect to Thue-Morse-Sturmian words.
This is joint work with Vilmos Komornik and Yuru Zou.
dynamical systemsnumber theory
Audience: researchers in the topic
( slides )
Series comments: Description: Online seminar on numeration systems and related topics
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Organizers: | Shigeki Akiyama, Ayreena Bakhtawar, Karma Dajani, Kevin Hare, Hajime Kaneko, Niels Langeveld, Lingmin Liao, Wolfgang Steiner* |
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