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arXiv · 1602.06173

On small univoque bases of real numbers

Derong Kong
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Abstract

Given a positive real number $x$, we consider the smallest base $q_s(x)\in(1,2)$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that \[ x=\sum_{i=1}^\infty\frac{d_i}{(q_s(x))^i}. \] In this paper we give complete characterizations of those $x$'s for which $q_s(x)\le q_{KL}$, where $q_{KL}$ is the Komornik-Loreti constant. Furthermore, we show that $q_s(x)=q_{KL}$ if and only if \[ x\in\left\{1, ~\frac{q_{KL}}{q_{KL}^2-1},~ \frac{1}{q_{KL}^2-1}, ~\frac{1}{q_{KL}(q_{KL}^2-1)}\right\}. \] Finally, we determine the explicit value of $q_s(x)$ if $q_s(x)<q_{KL}$.

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Derong Kong. 2016-05-23. On small univoque bases of real numbers. https://doi.org/10.1007/s10474-016-0637-7

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