Expansions in multiple bases (Q2036581)

After the seminal papers of \textit{A. Rényi} [Acta Math. Acad. Sci. Hung. 8, 477--493 (1957; Zbl 0079.08901)] and \textit{W. Parry} [Acta Math. Acad. Sci. Hung. 11, 401--416 (1960; Zbl 0099.28103)], many papers were devoted to the so-called ``beta-expansions'' of real numbers. The author of the paper under review studies expansions in multiple bases \((\beta_0, \beta_1, \ldots, \beta_m)\): if \(m\) is an integer and \(\beta_i > 1\) for all \(i\), the sequence \(w = (w_i) \in \{0, 1, \ldots, m\}^{\mathbb{N}}\) is a \((\beta_0, \beta_1, \ldots, \beta_m)\)-expansion of \(x\) if \[x = \sum_{i \geq 1} \frac{w_i}{\beta_{w_1}\beta_{w_2} \cdots \beta_{w_i}} \cdot \] This definition clearly generalizes the usual beta-expansion. The author proves several results that are generalizations of the corresponding results for beta-expansions, in particular about conditions on sequences \((w_i)\) to be \((\beta_0, \beta_1, \ldots, \beta_m)\)-expansions, and about greedy, lazy, quasi-greedy, quasi-lazy, and unique expansions, the situation being more precise in case \(m =1\) or \(m = 2\). Note that a recent preprint of \textit{Y. Zou}, \textit{V. Komornik} and \textit{J. Lu} [Expansions in multiple bases over general alphabets (\url{arxiv:2102.10051})] proposes a framework that generalizes both the definition above and the paper of Neunhäuserer. (Please also note that this paper of \textit{J. Neunhäuserer} has now appeared [Mediterr. J. Math. 18, No. 2, Paper No. 70, 8 p. (2021; Zbl 1468.11163)].)