Lattice points in the 3-dimensional torus

TL;DR

This paper establishes a new bound of 4/3 for the lattice point discrepancy in a 3D torus, advancing understanding of lattice point problems for complex solids.

Contribution

It proves a sharp discrepancy exponent for the 3D torus, extending classical exponential sum methods and applying to related solids in three-dimensional space.

Findings

Proves the exponent 4/3 for lattice point discrepancy in a 3D torus.

Identifies the limit of classical exponential sum methods for this problem.

Extends results to other solids related to the torus.

Abstract

We prove the exponent $4/3$ for the lattice point discrepancy of a torus in $\mathbb{R}^3$ (generated by the rotation of a circle around the $z$ axis). The exponent comes from a diagonal term and it seems a natural limit for any approach based solely on classical methods of exponential sums. The result extends to other solids in $\mathbb{R}^3$ related to the torus.