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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.14428 |
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Table of Contents:
- Given $β>1$ and $α\in[0,1)$, let $T_{β, α}(x)=βx+α\pmod 1$. Then under the map $T_{β,α}$ each $x\in[0,1]$ has an \emph{intermediate $β$-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 $T_{β,α}$ by considering the expected value $M_β(α)$ of the \emph{normalized errors} $(θ_{β,α}^n(x))_{n\geq 1}$, 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 $M_β(\cdot)$ is continuous on $[0,1)$. As a result, $\mathcal{M_β}:=\{M_β(α):α\in[0,1)\}$ is a closed interval. In particular, if $β$ is a multinacci number, the map $T_{β,α}$ has matching for Lebesgue almost every $α\in[0,1)$, and then $M_β(\cdot)$ is locally linear almost everywhere on $[0,1)$.