Covering three-tori with cubes

Abstract

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