On the lower bound in the lattice point remainder problem for a parallelepiped

Abstract

Let $ \Gamma \subset \RR^s $ be a lattice, obtained from a module in a totally real algebraic number field. Let $G$ be an axis parallel parallelepiped, and let $|G|$ be a volume of $G$. In this paper we prove that $\limsup_{|G| \to \infty} (\det \Gamma \#(\Gamma\cap G)-|G|)/\ln^{s-1} |G| >0.$ Thus the known estimate $\det \Gamma \#(\Gamma\cap G)=|G| +O(\ln^{s-1} |G|)$ is exact. We obtain also a similar result for the low discrepancy sequence corresponding to $\Gamma$.