Approximation properties of the intermediate $\beta$-expansions

Given $\beta >1$ and $\alpha \in [0,1)$, let $T_{\beta ,\alpha }(x)=\beta x+\alpha (\mathrm{m}\mathrm{o}\mathrm{d}1)$. Then under the map $T_{\beta ,\alpha }$ each $x\in [0,1]$ has an \emph{intermediate $\beta$-expansion} of the form $x={\sum }_{i=1}^{\mathrm{\infty }}\frac{c_i-\alpha }{{\beta }^i}$ {with each $c_i\in \{0,1,\dots ,\text{lf}\beta +\alpha \text{rf}\}$}. 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} $({\theta }_{\beta ,\alpha }^n(x){)}_{n\ge 1}$, where $${\theta }_{\beta ,\alpha }^n(x):={\beta }^n(x-\sum _{i=1}^n\frac{c_i-\alpha }{{\beta }^i}),n\in \mathbb{N}\mathrm{.}$$ We prove that $M_{\beta }(\cdot )$ is continuous on $[0,1)$. As a result, ${\mathcal{M}}_{\mathcal{\beta }}:=\{M_{\beta }(\alpha ):\alpha \in [0,1)\}$ 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 $[0,1)$.