## Comments on the distribution modulo one of powers of Pisot and Salem numbers.(English)Zbl 1263.11067

Define the norm $$\| t\|$$ of a real number $$t$$ by the distance between $$t$$ and the nearest integer.
Let $$\alpha$$ be an algebraic number greater than 1. Let $$M_{\alpha}(x) =x^d+(a_{d-1}/b_{d-1}) x^{d-1} +\dots +a_0/b_0$$ be the minimal polynomial of $$\alpha,$$ where $$(a_k,b_k)\in {\mathbb Z}\times {\mathbb N}$$ and gcd $$(a_k,b_k)=1,$$ and let $C(\alpha)=\bigl((1+\sum_{k=0}^{d-1}| a_k/b_k| ) \cdot \text{lcm}(b_0, b_1, \dots, b_{d-1}) \bigr)^{-1}.$ The main results of the paper are the following.
(i) If $$\limsup_{n\to \infty} \| \lambda\alpha^n\| <C(\alpha)$$ holds for some $$\lambda$$ then $$\alpha$$ is Salem or PV number and $$\lambda$$ belongs to the set $$\Lambda(\alpha)$$ of numbers of the form $$\beta /\alpha^p M'_{\alpha}(\alpha )$$, where $$p\in {\mathbb N}$$ and $$\beta \in {\mathbb Z}[\alpha]$$.
(ii) $$\limsup_{n\to \infty} \| \lambda\alpha^n\| =0$$ if and only if $$\alpha$$ is a PV number and $$\lambda \in \Lambda(\alpha)$$.
(iii) Let $$\alpha$$ be a Salem number and $$\epsilon \in ]0, C(\alpha)]$$. The number $$\lambda$$ satisfies $${\limsup}_{n\to \infty} \| \lambda\alpha^n\| <\varepsilon$$ if and only if $\lambda\in \{t\in \Lambda(\alpha),\;\sum_{i\in I}| \sigma_i(t)| <\varepsilon\},$ where $$\{\sigma_i(t)\}$$ is the set of nonreal embeddings of $${\mathbb Q}(\alpha)$$ into $${\mathbb C}$$.

### MSC:

 11J71 Distribution modulo one 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11R04 Algebraic numbers; rings of algebraic integers
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