Sums of Reciprocals of Fractional Parts over Aligned Boxes

TL;DR

This paper establishes new volume-dependent upper bounds for sums of reciprocals of fractional parts over aligned boxes, extending previous symmetric box results through innovative lattice-point counting methods.

Contribution

It introduces a novel lattice-point counting technique that yields bounds depending only on volume, generalizing prior symmetric box results to more general aligned boxes.

Findings

Upper bounds depend solely on box volume

Extension from symmetric to general aligned boxes

Uses advanced lattice-point counting methods

Abstract

In this paper, we prove new upper bounds for sums of reciprocals of fractional parts over general aligned boxes, thus extending a previous result of the author concerning bounds for sums of reciprocals over symmetric boxes. These new upper bounds depend solely on the volume of the boxes, and not on their diameter. This generalisation relies a novel lattice-point counting technique involving estimates for the higher successive minima of certain naturally arising lattices.