Let \( \Gamma \subset \mathbb{R}^s \) be a lattice obtained from a module in a totally real algebraic number field. Let \( \mathcal{R}(\theta, \mathbf{N}) \) be the error term in the lattice point problem for the parallelepiped \( [-\theta_1 N_1, \theta_1 N_1] \times \cdots \times [-\theta_s N_s, \theta_s N_s] \). In this paper, we prove that \( \mathcal{R}(\theta, \mathbf{N}) / \sigma(\mathcal{R}, \mathbf{N}) \) has a Gaussian limiting distribution as \( N \to \infty \), where \( \theta = (\theta_1, \ldots, \theta_s) \) is a uniformly distributed random variable in \( [0,1]^s \), \( N = N_1 \cdots N_s \) and \( \sigma(\mathcal{R}, \mathbf{N}) \asymp (\log N)^{(s-1)/2} \). We obtain also a similar result for the low discrepancy sequence corresponding to \( \Gamma \). The main tool is the S-unit theorem.