Approximation properties of the intermediate ${\hat{I}}^2$-expansions

Abstract

Given ${\hat{I}}^2>1$ and $\hat{I}\pm \in [0,1)$, let $T_{{\hat{I}}^2,\hat{I}\pm }(x)={\hat{I}}^{2}x+\hat{I}\pm (\mathrm{mod}1)$. Then under the map $T_{{\hat{I}}^2,\hat{I}\pm }$ each $x\in [0,1]$ has an \emph{intermediate ${\hat{I}}^2$-expansion} of the form $x=\sum_{i=1}^\infty\frac{c_i-α}{β^i}$ {with each $c_i\in\{0,1,\ldots,\lf β+α\rf\}$}. In this paper we study the approximation properties of $T_{{\hat{I}}^2,\hat{I}\pm }$ by considering the expected value $M_{\hat{I}}^2(\hat{I}\pm )$ of the \emph{normalized errors} $(\hat{I}{¸}_{{\hat{I}}^2,\hat{I}\pm }^n(x){)}_{n\ge 1}$, where $$θ_{β,α}^n(x):=β^n\left(x-\sum_{i=1}^n\frac{c_i-α}{β^i}\right),\quad n\in\mathbb{N}.$$ We prove that $M_{\hat{I}}^2(\cdot )$ is continuous on $[0,1)$. As a result, ${\mathcal{M}}_{\hat{\mathcal{I}}}^2:=\{M_{\hat{I}}^2(\hat{I}\pm ):\hat{I}\pm \in [0,1)\}$ is a closed interval. In particular, if ${\hat{I}}^2$ is a multinacci number, the map $T_{{\hat{I}}^2,\hat{I}\pm }$ has matching for Lebesgue almost every $\hat{I}\pm \in [0,1)$, and then $M_{\hat{I}}^2(\cdot )$ is locally linear almost everywhere on $[0,1)$.

35 page,5 figures