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On small univoque bases of real numbers

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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

$$x\in \Big\{1, \frac{q_{KL}}{q_{KL}^2-1}, \frac{1}{q_{KL}^2-1}, \frac{1}{q_{KL}(q_{KL}^2-1)}\Big\}.$$

Finally, we determine the explicit value of \({q_s(x)}\) if \({q_s(x)<{q_{KL}}}\).

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Authors and Affiliations

  1. School of Mathematical Science, Yangzhou University, Yangzhou, JiangSu, 225002, People’s Republic of China

    D. Kong

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  1. D. Kong
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Correspondence to D. Kong.

Additional information

The author was supported by NSFC No. 11401516 and Jiangsu Province Natural Science Foundation for the Youth No. BK20130433.

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Kong, D. On small univoque bases of real numbers. Acta Math. Hungar. 150, 194–208 (2016). https://doi.org/10.1007/s10474-016-0637-7

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  • DOI: https://doi.org/10.1007/s10474-016-0637-7

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