Approximation properties of intermediate $\beta $-expansions
Abstract
Given $\beta \gt 1$ 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 intermediate $\beta $-expansion of the form $x=\sum_{i=1}^\infty \frac{c_i-\alpha }{\beta ^i}$ with $c_i\in \{0,1,\ldots ,\lfloor \beta +\alpha \rfloor \}$ for each $i$. We study the approximation properties of $T_{\beta ,\alpha }$ by considering the expected value $M_\beta (\alpha )$ of the normalized errors $(\theta _{\beta ,\alpha }^n(x))_{n\geq 1}$, where $$\theta _{\beta ,\alpha }^n(x):=\beta ^n\bigg(x-\sum _{i=1}^n\frac {c_i-\alpha }{\beta ^i}\bigg),\quad n\in \mathbb {N}.$$ 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 Lebesgue almost everywhere on $[0,1)$.
