Perfectly packing a cube by cubes of nearly harmonic sidelength

TL;DR

This paper proves that for certain parameters, large collections of small cubes with nearly harmonic side lengths can perfectly pack a larger cube, extending previous results in specific dimensions and parameter ranges.

Contribution

It generalizes and improves upon prior packing results by establishing perfect packings of cubes with nearly harmonic sizes in higher dimensions for large n.

Findings

Perfect packing of d-dimensional cubes with side length n^{-t} for large n.

Extension of Tao's two-dimensional packing results to higher dimensions.

Improved bounds on the parameter t for which perfect packings are possible.

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 $n \geq n_0$ 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 \leq 4/11 $ and $n_0 = 1$ and builds upon recent work of Terence Tao in the two-dimensional case.