论文标题
数字表示的共同正态性：一种厄贡方法
Joint normality of representations of numbers: an ergodic approach
论文作者
论文摘要
我们引入了一种数字表示{\ em联合正态性}的研究。例如，我们表明，对于任何整数$ b \ geq 2 $，几乎每个数字$ x \ in [0,1）$在$ b $ expandion和持续的分数扩展方面是共同正常的。这个事实是以下结果的推论，涉及{\ em pointwisce intionwistion intergodicity}： 令$ t_b：[0,1] \ rightarrow [0,1] $ be times $ b $映射由$ t_b x = bx \，\ bmod \，1 $，让$ t_g：[0,1] \ rightarrow [0,1] $ be be the Gauss $由$ t_g（x）= $ t_g（x） \ ne 0 $和$ t_g（0）= 0。 \ sum_ {n = 0}^{n-1} f（t_b^{n} x）\，g（t_g^n x）= \ int f \，dλ\ cdot \ cdot \ cdot \ cd，dμ_g\dμ_g\ quad \ quad \ quad \ quad \ quad \ quad \ quat text {几乎每个} $μ_g$是$ [0,1] $上的高斯度量，由$μ_g（a）= \ frac {1} {\ log 2} \ int_a \ frac {1} {1+x} {1+x} dx $用于任何可测量的套件$ a \ subset $ a \ subset [0,1] $。 我们表明，对于间隔的多种数量理论图，发生了尖端的关节恐怖性现象，并得出与关节正态性有关的相应推论。 我们还建立了各种形式的正态性和共同正态性的等效性，以形成数字的表示，从而为经典正态性结果提供了一个一般框架。
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.
