Temporal Central Limit Theorem for a Multidimensional Adding Machine

Let \(p_1,\ldots,p_{s+1}\) be distinct primes and let \(T_{p_i}\) be the
von Neumann–Kakutani adding machine
\[
T_{p_i}(x)=\bigl(T_{p_i}(x_1),\ldots,T_{p_i}(x_s)\bigr).
\]

Let \(y_i\in(0,1)\) be a \(p_{s+1}\)-rational \((1\le i\le s)\),
\(I_{[0,y)}\) the indicator function of the box
\([0,y_1)\times\cdots\times[0,y_s)\), and we prove the following general limit theorem:

\[
\frac{\sum_{k=n+1}^{2n} I_{[0,y)}\!\left(T_{\mathbf{p}}^{\,k}(x)\right)
-2n y_1y_2\cdots y_s}
{\mathcal{H}_N(x)\,\log^{s/2}N}
\;\xrightarrow{\mathcal D}\;
\mathcal N(0,1),
\]

when \(n\) is sampled uniformly from \(\{1,\ldots,N\}\),
\(\mathcal H_N(x)\in[v_1,v_2]\) with some \(v_1,v_2>0\),
for almost all \(x\in[0,1)^s\). The main tool in the proof is the
\(S\)-unit theorem and the theorem on linear forms in the logarithm.