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_{ $P$ }( $x$ ) =(T_{p_1}(x_1),…, T_{p_s}(x_s)) $. L e t$ y_i \in (0,1) $b e a$ p_{s+1} $- r a t i o n a l$ (1 \leq i \leq s) $,$ $1$ {[0, $y$ )}$ the indicator function of the box $[0, y_{1}) \times \dots \times [0, y_{s})$ . In this paper, we prove the following central limit theorem: \begin{equation} \nonumber \frac{ \sum{k=-n}^{n-1} $1$ _{[0, $y$ )}(T^k_P( $x$ )) -2n y_1 y_2\dots y_s }{ $H$ _N( $x$ ) \log_2^{s/2} N} ; \stackrel{w}{\longrightarrow} ; $N$ (0,1), \end{equation} when $n$ is sampled uniformly from ${1, . . ., N}$ , $ $H$ _N( $x$ ) \in [\upsilon_1, \upsilon_2] $w i t h s o m e$ \upsilon_1, \upsilon_2 >0 $, f o r a l m o s t a l l$ $x$ \in [0,1)^s$.

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_{ $P$ }( $x$ ) =(T_{p_1}(x_1),…, T_{p_s}(x_s)) $. L e t$ y_i \in (0,1) $b e a$ p_{s+1} $- r a t i o n a l$ (1 \leq i \leq s) $,$ $1$ {[0, $y$ )}$ the indicator function of the box $[0, y_{1}) \times \dots \times [0, y_{s})$ . In this paper, we prove the following central limit theorem: \begin{equation} \nonumber \frac{ \sum{k=-n}^{n-1} $1$ _{[0, $y$ )}(T^k_P( $x$ )) -2n y_1 y_2\dots y_s }{ $H$ _N( $x$ ) \log_2^{s/2} N} ; \stackrel{w}{\longrightarrow} ; $N$ (0,1), \end{equation} when $n$ is sampled uniformly from ${1, . . ., N}$ , $ $H$ _N( $x$ ) \in [\upsilon_1, \upsilon_2] $w i t h s o m e$ \upsilon_1, \upsilon_2 >0 $, f o r a l m o s t a l l$ $x$ \in [0,1)^s$.

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