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

Let 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 lim sup |G|→∞ (det Γ#(Γ ∩G)− |G|)/ ln |G| > 0. Thus the known estimate det Γ#(Γ ∩ G) = |G| + O(ln |G|) is exact. We obtain also a similar result for the low discrepancy sequence corresponding to Γ.