Affiliation:
1. Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel
Abstract
Abstract
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 [−θ
1
N
1, θ
1
N
1] × . . . × [−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
= (θ
1, . . . , θs
) is a uniformly distributed random variable in [0, 1]
s
, N = N
1 . . . . 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.
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