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Journal

Uniform Distribution Theory (UDT)


Volume & Issue

UDT. Volume 20. Issue 1 & 2


Published on

December, 2025


Pages

178-223


DOI

http://doi.org/10.2478/UDT-2025-0010

Article

Temporal Central Limit Theorem for a Multidimensional Adding Machine


Authors

Mordechay B. Levin


Abstract

Let p1,...,ps+1 be distinct primes and let Tpi be the von Neumann-Kakutani adding machine (1 ≤ i ≤ s), T𝒫 (x)= (Tp1 (x1),...,Tps(xs)). Let yi ∈ (0, 1) be a ps+1-rational (1 ≤ i ≤ s), 𝟙[0,y) the indicator function of the box [0,y1)×…×[0,ys). 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)

when n is sampled uniformly from {1,...,N}, ℋN (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.


Citation

Levin, M.B.. (2025). Temporal Central Limit Theorem for a Multidimensional Adding Machine. Uniform distribution theory, 20(1), 178-223. https://doi.org/10.2478/udt-2025-0010