Approximation orders of the unit in the β-dynamical systems

Abstract

For any real number β > 1, let S _n (β) be the partial sum of the first n items of the β-expansion of 1. It was known that the approximation order of 1 by S _n (β) is β ⁻ⁿ for Lebesgue almost all β > 1. We consider the size of the set of β > 1 for which 1 can be approximated with the other orders ${\beta^{-\varphi(n)}}$ , where ${\varphi}$ is a positive function defined on ${\mathbb N}$ . More precisely, the size of the sets

$$\left\{\beta\in \mathfrak{B}:\limsup_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$

and

$$\left\{\beta\in \mathfrak{B}:\liminf_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$

are determined, where ${\mathfrak{B}=\{ \beta>1:\beta \text{ is not a simple Parry number}\}}$ .