On the Gaussian limiting distribution of lattice points in a parallelepiped

Let Γ⊂ Rs 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]×···×[− θsNs, θsNs]. In this paper, we prove that R (θ, N)/σ (R, N) has a Gaussian limiting distribution as N→∞, where θ=(θ1,..., θs) is a uniformly distributed random variable in [0, 1] s, N= N1··· Ns 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