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Diophantine analysis of the expansions of a fixed point under continuum many bases

In this paper, we study the Diophantine properties of the orbits of a fixed point in its expansions under continuum many bases. More precisely, let $T_β$ be the beta-transformation with base $β>1$, $\{x_{n}\}_{n\geq 1}$ be a sequence of real numbers in $[0,1]$ and $φ\colon \mathbb{N}\rightarrow (0,1]$ be a positive function. With a detailed analysis on the distribution of {\em full cylinders} in the base space $\{β>1\}$, it is shown that for any given $x\in(0,1]$, for almost all or almost no bases $β>1$, the orbit of $x$ under $T_β$ can $φ$-well approximate the sequence $\{x_{n}\}_{n\geq 1}$ according to the divergence or convergence of the series $\sum φ(n)$. This strengthens Schmeling's result significantly and complete all known results in this aspect. Moreover, the idea presented here can also be used to determine the Lebesgue measure of the set \begin{equation*} \{x\in [0,1]\colon|T^{n}_βx-L(x)|<φ(n) \text{ for infinitely many } n\in\mathbb{N}\}, \end{equation*} for a fixed base $β>1$, where $L\colon [0,1]\rightarrow[0,1]$ is a Lipschitz function.