Beta-expansions for infinite families of Pisot and Salem numbers
In \(\beta\)-expansions of real numbers the bases \(\beta \in (1,2)\) that are Pisot numbers or Salem numbers play an important role. It is known that if the expansion of \(x \in {\mathbb Q} \cap [0,1)\) is periodic or finite then \(\beta\) must be a Pisot number or a Salem number. The reverse implication for Salem numbers is an open problem. In this paper the authors study some \(\beta\)-expansions and greedy expansions for some special infinite families of Pisot and Salem numbers \(\beta\). For example, \(\beta\)-expansion of the Pisot number defined by \(\Phi_r(x)=x^{r+1}-2x^r+x-1\), \(r \in {\mathbb N}\), is given by \(1^r0^{r-1}1\). The polynomial \(x^k \Phi_r(x)-\Phi_r^{*}(x)\), \(k \in {\mathbb N}\), defines a Salem number. Their greedy expansions are established for \(r \geq k\) and \(r-k\) not divisible by \(2k\). The authors remark that the case \(k=3\), \(r=9\), where \(2k|(r-k)\), leads to an open problem, probably a non-periodic \(\beta\)-expansion.
scientific article; zbMATH DE number 5368241
scientific article