Approximation properties of the intermediate $\beta$-expansions

Given $\beta>1$ and $\alpha\in[0,1)$, let $T_{\beta, \alpha}(x)=\beta x+\alpha\pmod 1$. Then under the map $T_{\beta,\alpha}$ each $x\in[0,1]$ has an \emph{intermediate $\beta$-expansion} of the form $x=\sum_{i=1}^\infty\frac{c_i-\alpha}{\beta^i}$ {with each $c_i\in\{0,1,\ldots,\lf \beta+\alpha\rf\}$}. In this paper we study the approximation properties of $T_{\beta,\alpha}$ by considering the expected value $M_\beta(\alpha)$ of the \emph{normalized errors} $(\theta_{\beta,\alpha}^n(x))_{n\geq 1}$, where $$\theta_{\beta,\alpha}^n(x):=\beta^n\left(x-\sum_{i=1}^n\frac{c_i-\alpha}{\beta^i}\right),\quad n\in\mathbb{N}.$$ We prove that $M_\beta(\cdot)$ is continuous on $[0,1)$. As a result, $\mathcal{M_\beta}:=\{M_\beta(\alpha):\alpha\in[0,1)\}$ is a closed interval. In particular, if $\beta$ is a multinacci number, the map $T_{\beta,\alpha}$ has matching for Lebesgue almost every $\alpha\in[0,1)$, and then $M_\beta(\cdot)$ is locally linear almost everywhere on $[0,1)$.