Decomposition of a cube into nearly equal smaller cubes

Abstract

Let $d$ be a fixed positive integer and let $\epsilon >0$. It is shown that for every sufficiently large $n\ge n_0(d,\epsilon )$, the $d$-dimensional unit cube can be decomposed into exactly $n$ smaller cubes such that the ratio of the side length of the largest cube to the side length of the smallest one is at most $1+\epsilon$. Moreover, for every $n\ge n_0$, there is a decomposition with the required properties, using cubes of at most $d+2$ different side lengths. If we drop the condition that the side lengths of the cubes must be roughly equal, it is sufficient to use cubes of two different sizes.

13 pages, 4 figures