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

TL;DR

This paper establishes a lower bound for the lattice point remainder problem in parallelepipeds, confirming the sharpness of known estimates and extending results to low discrepancy sequences linked to such lattices.

Contribution

It proves a new lower bound for the lattice point remainder problem in parallelepipeds derived from algebraic number fields, confirming the optimality of existing upper bounds.

Findings

Lower bound for lattice point discrepancy is strictly positive.

Known estimate for lattice point count is sharp and cannot be improved.

Results extend to low discrepancy sequences associated with the lattice.

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 $\Gamma$.