Perfectly packing a cube by cubes of nearly harmonic sidelength

Abstract

Let $d$ be an integer greater than $1$, and let $t$ be fixed such that $\frac{1}{d}<t<\frac{1}{d-1}$. We prove that for any $n_0$ chosen sufficiently large depending upon $t$, the $d$-dimensional cubes of sidelength $n^{-t}$ for $n\ge n_0$ can perfectly pack a cube of volume ${\sum }_{n=n_0}^{\infty }\frac{1}{n^{dt}}$. Our work improves upon a previously known result in the three-dimensional case for when $1/3<t\le 4/11$ and $n_0=1$ and builds upon recent work of Terence Tao in the two-dimensional case.