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] × . . . × [−θs Ns, θs Ns]. 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.

The hyperbolic lattice point problem in conjugacy classes

For Γ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces ${{\Gamma\backslash{\mathbb{H}}}}$ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound ${O(X^{2/3})}$, due to Good. For ${{\mathrm{SL}_{2}({\mathbb{Z}})}}$ we interpret our results in terms of indefinite quadratic forms.