On the Lower Bound in the Lattice Point Remainder Problem for a Parallelepiped

Abstract

Let $$ \Gamma \subset {\mathbb {R}}^s $$Γ?Rs 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 [Equation not available: see fulltext.]Thus the known estimate $$\det \Gamma \#(\Gamma \cap G)=|G| +O(\ln ^{s-1} |G|)$$detΓ#(Γ?G)=|G|+O(lns-1|G|) is exact. We obtain also a similar result for the low discrepancy sequence corresponding to $$\Gamma $$Γ.