Unique double base expansions (Q6565881)

Suppose \(\beta>1\) is a fixed real number. The present research deals with a certain generalization for the classical \(\beta\)-expansions, which were introduced by \textit{A. Rényi} [Acta Math. Acad. Sci. Hung. 8, 477--493 (1957; Zbl 0079.08901)].\N\NIn this paper, the following expansions (\((q_{0}, q_{1})\)-expansions) are considered: \N\[\N\sum_{k = 1}^{\infty} {\frac{i_{k}}{q_{i_{1}}q_{i_{2}} \dots q_{i_{k}}}}, \N\]\Nwhere \(q_{0}, q_{1} > 1\) are fixed real numbers and \(i_{k} \in \{0, 1\}\). Also, some attention is given to certain cases of the following expansion with more general alphabets \(\{d_0, d_1,\dots , d_m\}\) with real digits: \N\[\N\sum_{k = 1}^{\infty} {\frac{d_{i_{k}}}{q_{i_{1}}q_{i_{2}} \dots q_{i_{k}}}}, \N\]\Nwhere \(m\) is a positive integer, \(i_k\in \{0,1, \dots , m\}\), and \(q_0, q_1, \dots , q_m >1\).\N\NIt is noted that one of the main goals is to extend results on the cardinality of univoque sets to alphabet-base systems.\N\NSome functions and sets, which are related to the considered expansion of numbers, are the main objects of this investigation.\N\NThis research also contains detail explanations of used notions and a brief survey on related results. Proofs are given with auxiliary descriptions, figures, and statements. Finally, several open problems were formulated.