Approximation properties of the intermediate $\beta$-expansions

TL;DR

This paper investigates the approximation properties of intermediate beta-expansions generated by a family of transformations, proving continuity of the expected normalized errors and analyzing their behavior for multinacci numbers.

Contribution

It establishes the continuity of the expected error function and characterizes the approximation properties of intermediate beta-expansions, especially for multinacci numbers.

Findings

The expected error function $M_eta(eta)$ is continuous on [0,1).

The set of expected errors forms a closed interval.

For multinacci $eta$, the map exhibits matching for almost every $eta$.

Abstract

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 $[0,1)$.