Representations of a number in an arbitrary base with unbounded digits (Q6620599)

For \(\alpha,\beta\in \mathbb{C}\), let \(P_\beta(\alpha)\) denote the number of distinct representations of \(\alpha\) of the form\N\[\N\alpha= d_n\beta^n+d_{n-1}\beta^{n-1}+\cdots+d_1\beta+d_0,\N\]\Nwhere \(n,d_0,d_1,\ldots,d_n\) are nonnegative integers. The main goal of the paper under review is to give a characterization of all complex numbers \(\beta\) which have the following property:\N\begin{itemize}\N\item[(i)] \(P_\beta(\alpha)<\infty\) for all \(\alpha\in \mathbb{C}\).\N\end{itemize}\N\NIt is proven that a necessary and sufficient condition for (i) is\N\begin{itemize}\N\item[(ii)] \(\beta\) is either a transcendental number or an algebraic number which has a conjugate over \(\mathbb{Q}\) lying in the interval \((1,\infty)\).\N\end{itemize}\NIt can be deduced easily from some observations that (ii) implies (i) and that (i) does not hold for every rational number \(\beta\in (-\infty, 1]\). The key result of this paper is a proof of the remaining part: for each non-rational algebraic number \(\beta\) whose conjugates are all in \(\mathbb{C}\backslash(1,\infty)\), there exists \(\alpha\in \mathbb{C}\) for which \(P_\beta(\alpha)=\infty\). In fact, by using some known results, also proven by the author, it can be shown that if \(\beta\) is an algebraic numbers whose conjugates are all in \(\mathbb{C}\backslash(0,\infty)\), then \(P_\beta(0)=\infty\). Hence what actually remains to consider is the case when \(\beta\) has all conjugates in \((0,1)\). The proof relies on some auxiliary lemmas and simple, yet elegant, algebraic arguments.