## On the spectra of Pisot numbers.(English)Zbl 1303.11118

Summary: Let $$\theta$$ be a real number greater than 1, and let $$((\cdot))$$ be the fractional part function. Then, $$\theta$$ is said to be a $$Z$$-number if there is a non-zero real number $$\lambda$$ such that $$((\lambda\theta^n)) < \tfrac 12$$ for all $$n\in\mathbb N$$. A. Dubickas [Glasg. Math. J. 48, No. 2, 331–336 (2006; Zbl 1138.11026)] showed that strong Pisot numbers are $$Z$$-numbers. Here it is proved that $$\theta$$ is a strong Pisot number if and only if there exists $$\lambda\neq 0$$ such that $$((\lambda \alpha )) < \tfrac 12$$ for all $$\alpha\in\left\{\theta^n\mid n\in \mathbb{N}\right\} \cup \left\{\sum_{n=0}^N\theta^n\mid N\in\mathbb{N}\right\}$$. Also, the following characterisation of Pisot numbers among real numbers greater than 1 is shown: $$\theta$$ is a Pisot number $$\Leftrightarrow\exists\lambda \neq 0$$ such that $$\| \lambda \alpha \| <\tfrac 13$$ for all $$\alpha\in\left\{\sum_{n=0}^N a_n\theta^n\mid a_n\in \{0,1\}, N\in \mathbb N\right\}$$, where $$\| \lambda \alpha \| = \min\{((\lambda \alpha)), 1- ((\lambda \alpha ))\}$$.

### MSC:

 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11J71 Distribution modulo one

### Keywords:

strong Pisot numbers

Zbl 1138.11026
Full Text:

### References:

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