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

Abstract

Let $ Γ\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 ${\mathrm{limsup}}_{|G|\to \infty }(\det \Gamma \#(\Gamma \cap G)-|G|)/{\ln }^{s-1}|G|>0.$ 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 $\Gamma$.

Cited by in corpus (1)

• On the lower bound of the discrepancy of Halton's sequence