Temporal Central Limit Theorem for a Multidimensional Adding Machine

Let p₁,...,p_s+1 be distinct primes and let T_pi be the von Neumann-Kakutani adding machine (1 ≤ i ≤ s), T_𝒫 (x)= (T_p1 (x₁),...,T_ps(x_s)). Let y_i ∈ (0, 1) be a p_s+1-rational (1 ≤ i ≤ s), 𝟙_[0,y) the indicator function of the box [0,y₁)×…×[0,y_s). In this paper, we prove the following central limit theorem: ∑k=-nn-1𝟙[0,y)(TPk(x)-2ny1y2⋯ys)ℋN(x)⋅log2s/2N→N(0,1) \frac{\sum_{k=-n}^{\,n-1}\mathbb{1}_{[0,y)}\!\left(T_{P}^{k}(\mathbf{x})- 2n\,y_{1}y_{2}\cdots y_{s}\right)}{\mathcal{H}_{N}(\mathbf{x})\,\log_{2}^{\,s/2} N}\xrightarrow{w} \mathcal{N}(0,1) when n is sampled uniformly from {1,...,N}, ℋ_N (x)∈[υ₁,υ₂] with some υ₁,υ₂ > 0, for almost all x ∈[0, 1)_s. The main tool in the proof is the S-unit theorem and the theorem on linear forms in the logarithm.