  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
$T_{\beta}$ be the beta-transformation with base $\beta>1$, $\{x_{n}\}_{n\geq
1}$ be a sequence of real numbers in $[0,1]$ and $\varphi\colon
\mathbb{N}\rightarrow (0,1]$ be a positive function. With a detailed analysis
on the distribution of {\em full cylinders} in the base space $\{\beta>1\}$, it
is shown that for any given $x\in(0,1]$, for almost all or almost no bases
$\beta>1$, the orbit of $x$ under $T_{\beta}$ can $\varphi$-well approximate
the sequence $\{x_{n}\}_{n\geq 1}$ according to the divergence or convergence
of the series $\sum \varphi(n)$. 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}_{\beta}x-L(x)|<\varphi(n) \text{ for infinitely many
} n\in\mathbb{N}\}, \end{equation*} for a fixed base $\beta>1$, where $L\colon
[0,1]\rightarrow[0,1]$ is a Lipschitz function.
