Let $p_1,...,p_{ s+1}$ be distinct primes and let $T_{ p_i}$ be the von Niemann - Kakutani adding machine $(1 \leq i \leq s)$, $T_{ \mathcal{ P}}(\mathbf{ x}) =(T_{ p_1}(x_1),..., T_{ p_s}(x_s))$. Let $y_i \in (0,1)$ be a $p_{ s+1}$-rational $(1 \leq i \leq s)$, $\mathbf{ 1}_{ [0,\mathbf{ y})}$ the indicator function of the box $[0,y_1) \times \cdots\times [0,y_s)$. In this paper, we prove the following central limit theorem: \begin{ equation} \nonumber \frac{ \sum_{ k=-n}^{ n-1} \mathbf{ 1}_{ [0,\mathbf{ y})}(T^k_P(\mathbf{ x})) -2n y_1 y_2\dots y_s }{ \mathcal{ H}_N(\mathbf{ x}) \log_2^{ s/2} N} \; \stackrel{ w}{ \longrightarrow} \;\mathcal{ N}(0,1), \end{ equation} when $n$ is sampled uniformly from $\{ 1,...,N\}$, $\mathcal{ H}_N(\mathbf{ x}) \in [\upsilon_1, \upsilon_2]$ with some $\upsilon_1, \upsilon_2 >0$, for almost all $\mathbf{ x} \in [0,1)^s$.
Categories: math.NT math.DS