Periodicity and pure periodicity in alternate base systems (Q6554730)

The authors investigate periodicity of expansions in alternate base systems. A Cantor real base system is a representation of real numbers \(x\in[0, 1)\) as\N\[\Nx = \sum_{k=1}^\infty \frac{x_k}{\prod_{i=1}^k\beta_i}\N\]\Nwith \(x_k\in\mathbb{N}\) where \(\mathcal{B}=(\beta_i)_{i\ge 1}\) with \(\beta_i>1\) for all \(i\). If \(\mathcal{B}\) is purely periodic with period of length \(p\), \(\mathcal{B}\) is called an alternate base and we write \(\mathcal{B}=(\beta_1,\ldots,\beta_p)\). The quantity \(\delta= \prod_{i=1}^p \beta_i\) will occur frequently in the results of this paper.\N\NThe set \(\operatorname{Per}(\mathcal{B})\) is defined to be the set of numbers in the unit interval \([0, 1)\) with periodic expansion in the alternate base number system defined by \(\mathcal{B}\). An alternate base \(\mathcal{B}\) is said to satisfy the pure periodicity property if there exists an interval \([0, \gamma)\) with \(0<\gamma\le 1\) such that every rational in \([0, \gamma)\) has a purely periodic \(\mathcal{B}\) expansion. Finally, we say that \(\mathcal{B}\) has the finiteness property if adding and subtracting finite \(\mathcal{B}\) expansions again yields finite \(\mathcal{B}\) expansions.\N\NExtending previous results and proving a conjecture by \textit{É. Charlier} et al. [J. Number Theory 254, 184--198 (2024; Zbl 1536.11120)], the authors show that for an alternate base \(\mathcal{B}\) if \(\mathbb{Q}\cap [0, 1)\subseteq \operatorname{Per}(\mathcal{B})\), then \(\delta\) is either a Pisot or a Salem number and \(\beta_1\), \dots, \(\beta_p\in\mathbb{Q}(\delta)\).\N\NNext, the authors prove that if \(\mathcal{B}\) is an alternate base with the pure periodicity property, then \(\delta\) is a Pisot or a Salem unit, \(\beta_1\), \dots, \(\beta_p\in\mathbb{Q}(\delta)\), and the vector \((\psi(\beta_1), \ldots, \psi(\beta_p))\) is not positive for any non-identical embedding \(\psi\colon \mathbb{Q}(\delta)\to\mathbb{C}\).\N\NFinally, the authors prove a partial converse: if \(\mathcal{B}\) is an alternate base with the finiteness property such that \(\delta\) is a Pisot unit, then \(\mathcal{B}\) as well as all of its cyclic shifts satisfy the pure periodicity property.