二进制数和函数相关测度的渐近性质
On the asymptotic behaviour of the correlation measure of sum-of-digits function in base 2
摘要：
令$s\_2(x)$表示任何$x \in \mathbb{N}$的二进制扩展中的位数“$1$“。本文研究了$s\_2(x+a)-s\_2(x)$的平均分布$μ\_a$，证明了方程$$ s\_2(x+a)-s\_2(x)= d $$的解是由$\{0,1\}^*$中的有限前缀集唯一确定的，而$d$的差的概率分布是由系数为$l^1(\mathbb{Z})$的算子的矩阵的无穷乘积给出的.然后，用$l(a)$表示$a$的二进制展开式中的模式数$01$，我们给出了当$l(a)$趋于无穷大时该概率分布的渐近性质以及概率测度$μ\_a$的方差估计
英文摘要：
Let $s\_2(x)$ denote the number of digits "$1$" in a binary expansion of any $x \in \mathbb{N}$. We study the mean distribution $μ\_a$ of the quantity $s\_2(x+a)-s\_2(x)$ for a fixed positive integer $a$.It is shown that solutions of the equation$$ s\_2(x+a)-s\_2(x)= d $$are uniquely identified by a finite set of prefixes in $\{0,1\}^*$, and that the probability distribution of differences $d$ is given by an infinite product of matrices whose coefficients are operators of $l^1(\mathbb{Z})$.Then, denoting by $l(a)$ the number of patterns "$01$" in the binary expansion of $a$, we give the asymptotic behaviour of this probability distribution as $l(a)$ goes to infinity as well as estimates of the variance of the probability measure $μ\_a$
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