Limit laws in the lattice problem. IV. The special case of ℤ^d

2022/11/05 by Julien Trevisan, Trevisan, Julien
Mathematics · #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2211.02873

openalex publication_date 2022/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the error of the number of points of the lattice ℤ^d that fall into a dilated and translated hypercube centred around 0 and whose axis are parallel to the axis of coordinates. We show that if t, the factor of dilatation, is distributed according to the probability measure (1)/(T) ρ((t)/(T)) dt with ρ being a probability density over [0,1] the error, when normalized by t^d-1, converges in law when T → ∞ in the case where the translation is of the form X=(x,⋯,x) and in the case where the coordinates of X are independent between them, independent from t and distributed according to the uniform law over [-(1)/(2),(1)/(2)]. In both cases, we compute the characteristic function of the limit law.

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