Topology of univoque sets in double-base expansions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916714603610112 |
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| author | Komornik, Vilmos Li, Yichang Zou, Yuru |
| author_facet | Komornik, Vilmos Li, Yichang Zou, Yuru |
| contents | Given two real numbers $q_0,q_1>1$ satisfying $q_0+q_1\geq q_0q_1$ and two real numbers $d_0\ne d_1$, by a {double-base expansion} of a real number $x$ we mean a sequence $(i_k)\in \{0,1\}^{\infty}$ such that \begin{equation*} x=\sum_{k=1}^{\infty}\frac{d_{i_k}}{q_{i_1}q_{i_2}\cdots q_{i_k}}. \end{equation*} We denote by $\mathcal{U}_{q_0,q_1}$ the set of numbers $x$ having a unique expansion. The topological properties of $\mathcal{U}_{q_0,q_1}$ have been investigated in the equal-base case $q_0=q_1$ for a long time. We extend this research to the case $q_0\neq q_1$. While many results remain valid, a great number of new phenomena appear due to the increased complexity of double-base expansions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_21374 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Topology of univoque sets in double-base expansions Komornik, Vilmos Li, Yichang Zou, Yuru Dynamical Systems Number Theory 11A63, 11B83, 37B10, 68R15 Given two real numbers $q_0,q_1>1$ satisfying $q_0+q_1\geq q_0q_1$ and two real numbers $d_0\ne d_1$, by a {double-base expansion} of a real number $x$ we mean a sequence $(i_k)\in \{0,1\}^{\infty}$ such that \begin{equation*} x=\sum_{k=1}^{\infty}\frac{d_{i_k}}{q_{i_1}q_{i_2}\cdots q_{i_k}}. \end{equation*} We denote by $\mathcal{U}_{q_0,q_1}$ the set of numbers $x$ having a unique expansion. The topological properties of $\mathcal{U}_{q_0,q_1}$ have been investigated in the equal-base case $q_0=q_1$ for a long time. We extend this research to the case $q_0\neq q_1$. While many results remain valid, a great number of new phenomena appear due to the increased complexity of double-base expansions. |
| title | Topology of univoque sets in double-base expansions |
| topic | Dynamical Systems Number Theory 11A63, 11B83, 37B10, 68R15 |
| url | https://arxiv.org/abs/2504.21374 |