On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

Let Γ ⊂ ℝ^s be a lattice obtained from a module in a totally real algebraic number field. Let ℛ(θ, N) be the error term in the lattice point problem for the parallelepiped [−θ₁N₁, θ₁N₁] × . . . × [−θ_s N_s, θ_s N_s]. In this paper, we prove that ℛ(θ, N)/σ(ℛ, N) has a Gaussian limiting distribution as N→∞, where θ = (θ₁, . . . , θ_s) is a uniformly distributed random variable in [0, 1]^s, N = N₁ . . . . N_s and σ(ℛ, N) ≍ (log N)^(s−1)/2. We obtain also a similar result for the low discrepancy sequence corresponding to Γ. The main tool is the S-unit theorem.