Temporal Central Limit Theorem for Multidimensional Adding Machine

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 $\mathbf{x} \in [0,1)^s$.