## Arithmetical properties of real numbers related to beta-expansions.(English)Zbl 1443.11147

Summary: The main purpose of this paper is to study the arithmetical properties of values $$\sum_{m=0}^{\infty} \beta^{-w(m)}$$, where $$\beta$$ is a fixed Pisot or Salem number and $$w(m)$$ ($$m=0,1,\ldots$$) are distinct sequences of nonnegative integers with $$w(m+1)>w(m)$$ for any sufficiently large $$m$$. We first introduce the algebraic independence results of such values.
Our results are applicable to certain sequences $$w(m)$$ ($$m=0,1,\ldots$$) with $$\lim_{m\to\infty}w(m+1)/w(m)=1.$$ For example, we prove that two numbers $\sum_{m=1}^{\infty}\beta^{-\lfloor \varphi(m)\rfloor}, \quad \sum_{m=3}^{\infty}\beta^{-\lfloor a(m)\rfloor}$ are algebraically independent, where $$\varphi(m)=m^{\log m}$$ and $$a(m)=m^{\log\log m}$$. Moreover, we also give the linear independence results of real numbers.
Our results are applicable to the values $$\sum_{m=0}^{\infty}\beta^{-\lfloor m^\rho\rfloor}$$, where $$\beta$$ is a Pisot or Salem number and $$\rho$$ is a real number greater than 1.

### MSC:

 11J91 Transcendence theory of other special functions 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11J72 Irrationality; linear independence over a field 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
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