Covering three-tori with cubes

TL;DR

This paper investigates the minimum number of cubes needed to cover the three-dimensional torus, establishing new bounds and exact values for specific cube sizes, advancing understanding of geometric coverings in topology.

Contribution

It provides new lower and upper bounds for the covering number and determines exact values for a range of cube sizes on the three-torus.

Findings

Established new bounds for (5)

Derived exact covering numbers for specific  ranges

Extended understanding of geometric coverings in topology

Abstract

Let $\mu(\varepsilon)$ be the minimum number of cubes of side $\varepsilon$ needed to cover the unit three-torus $[\mathbb{R}/\mathbb{Z}]^3$. We prove new lower and upper bounds for $\mu(\varepsilon)$ and find the exact value for all $\varepsilon\geq\frac{7}{15}$ and all $\varepsilon\in\left[\frac{1}{r+1/(r^2+r+1)},\frac{1}{r-1/(r^2-1)}\right)$ for any integer $r\geq 2$.