Approximation orders of the unit in the beta-dynamical systems
Approximation orders of the unit in the beta-dynamical systems
复制标题
β-动力系统中单位的近似阶数
DOI：
10.1007/s10474-017-0776-5
复制
发表时间：
2018
影响因子：
0.9
通讯作者：
Chen Y H
中科院分区：
文献类型：
--
作者：
Cao C Y;Chen Y H
For any real numberβ> 1, letSn(β) be the partial sum of the firstnitems of theβ-expansion of 1. It was known that the approximation order of 1 bySn(β) isβ−nfor Lebesgue almost allβ> 1. We consider the size of the set ofβ> 1 for which 1 can be approximated with the other orders $${\beta^{-\varphi(n)}}$$ β - φ ( n ) , where $${\varphi}$$ φ is a positive function defined on $${\mathbb N}$$ N . More precisely, the size of the sets $$\left\{\beta\in \mathfrak{B}:\limsup_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$ β ∈ B : lim sup n → ∞ log β ( 1 - S n ( β ) ) φ ( n ) = - 1 and $$\left\{\beta\in \mathfrak{B}:\liminf_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$ β ∈ B : lim inf n → ∞ log β ( 1 - S n ( β ) ) φ ( n ) = - 1 are determined, where $${\mathfrak{B}=\{ \beta>1:\beta \text{ is not a simple Parry number}\}}$$ B = { β > 1 : β is not a simple Parry number } .
For any real numberβ> 1, letSn(β) be the partial sum of the firstnitems of theβ-expansion of 1. It was known that the approximation order of 1 bySn(β) isβ−nfor Lebesgue almost allβ> 1. We consider the size of the set ofβ> 1 for which 1 can be approximated with the other orders $${\beta^{-\varphi(n)}}$$ β - φ ( n ) , where $${\varphi}$$ φ is a positive function defined on $${\mathbb N}$$ N . More precisely, the size of the sets $$\left\{\beta\in \mathfrak{B}:\limsup_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$ β ∈ B : lim sup n → ∞ log β ( 1 - S n ( β ) ) φ ( n ) = - 1 and $$\left\{\beta\in \mathfrak{B}:\liminf_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$ β ∈ B : lim inf n → ∞ log β ( 1 - S n ( β ) ) φ ( n ) = - 1 are determined, where $${\mathfrak{B}=\{ \beta>1:\beta \text{ is not a simple Parry number}\}}$$ B = { β > 1 : β is not a simple Parry number } .