Comments on the distribution modulo one of powers of Pisot and Salem numbers (Q2915408)

Define the norm \(\| t\| \) of a real number \(t\) by the distance between \(t\) and the nearest integer.NEWLINENEWLINELet \(\alpha\) 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 NEWLINE\[NEWLINE C(\alpha)=\bigl((1+\sum_{k=0}^{d-1}| a_k/b_k| ) \cdot \text{lcm}(b_0, b_1, \dots, b_{d-1}) \bigr)^{-1}.NEWLINE\]NEWLINE The main results of the paper are the following.NEWLINENEWLINE (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 \(\beta \in {\mathbb Z}[\alpha]\).NEWLINENEWLINE (ii) \(\limsup_{n\to \infty} \| \lambda\alpha^n\| =0 \) if and only if \(\alpha\) is a PV number and \(\lambda \in \Lambda(\alpha)\).NEWLINENEWLINE (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 NEWLINE\[NEWLINE \lambda\in \{t\in \Lambda(\alpha),\;\sum_{i\in I}| \sigma_i(t)| <\varepsilon\}, NEWLINE\]NEWLINE where \(\{\sigma_i(t)\}\) is the set of nonreal embeddings of \({\mathbb Q}(\alpha)\) into \({\mathbb C}\).