数的表示的联合正规性：一种遍历方法
Joint normality of representations of numbers: an ergodic approach
AI总结：
本研究提出遍历方法研究数的表示的联合正规性，证明逐点联合遍历性对多种区间数论映射成立，推导联合正规性推论，并建立多种正规性形式的等价性，提供通用研究框架。
摘要：
我们提出了一种遍历方法，用于研究数的表示的**联合正规性**。例如，我们证明了对任意整数$b\geq2$，几乎所有$x\in[0,1)$都关于$b$进制展开与连分数展开具有联合正规性。这一结论是下述**逐点联合遍历性**结果的推论：设$T_b:[0,1]\rightarrow[0,1]$为乘$b$映射，定义为$T_b x = bx \bmod 1$；设$T_G:[0,1]\rightarrow[0,1]$为高斯映射，定义为$x\neq0$时$T_G(x)=\{\frac{1}{x}\}$，$T_G(0)=0$（其中$\{\cdot\}$表示取小数部分）。对任意$f,g\in L^{\infty}(\lambda)$，有\\[ \lim_{N \rightarrow \infty} \frac{1}{N } \sum_{n=0}^{N-1} f(T_b^{n}x) \\, g(T_G^n x) = \int f \\, d \lambda\cdot \int g \\, d \mu_G \quad \text{对几乎所有 } x \in [0,1], \\]其中$\lambda$是$[0,1]$上的勒贝格测度，$\mu_G$是$[0,1]$上的高斯测度，对任意可测集$A\subset[0,1]$，$\mu_G(A)=\frac{1}{\log 2}\int_A\frac{1}{1+x}dx$。我们证明逐点联合遍历性现象在区间上的各类数论映射中均成立，并推导了对应的联合正规性推论。我们还建立了数的表示的多种正规性与联合正规性形式之间的等价性，从而为经典正规性结果提供了一个通用框架。
英文摘要：
We introduce an ergodic approach to the study of {\em joint normality} of representations of numbers. For example, we show that for any integer $b \geq 2$ almost every number $x \in [0,1)$ is jointly normal with respect to the $b$-expansion and continued fraction expansion. This fact is a corollary of the following result which deals with {\em pointwise joint ergodicity}: Let $T_b:[0,1] \rightarrow [0,1]$ be the times $b$ map defined by $T_b x = bx \, \bmod \, 1 $ and let $T_G:[0,1] \rightarrow [0,1]$ be the Gauss map defined by $T_G(x) = \{\frac{1}{x}\}$ for $x \ne 0$ and $T_G (0) =0.$ (Here $\{ \cdot \}$ denotes the fractional part.) For any $f, g \in L^{\infty} (λ)$, \[ \lim_{N \rightarrow \infty} \frac{1}{N } \sum_{n=0}^{N-1} f(T_b^{n}x) \, g(T_G^n x) = \int f \, d λ\cdot \int g \, d μ_G \quad \text{for almost every } x \in [0,1], \] where $λ$ is the Lebesgue measure on $[0,1]$ and $μ_G$ is the Gauss measure on $[0,1]$ given by $μ_G (A) = \frac{1}{ \log 2} \int_A \frac{1}{1+x} dx$ for any measurable set $A \subset [0,1]$. We show that the phenomenon of the pointwise joint ergodicity takes place for a wide variety of number-theoretical maps of the interval and derive the corresponding corollaries pertaining to joint normality. We also establish the equivalence of various forms of normality and joint normality for representations of numbers, hereby providing a general framework for classical normality results.
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