Decomposition of a cube into nearly equal smaller cubes

Let $d$ be a fixed positive integer and let $ε>0$. It is shown that for every sufficiently large $n\geq n_0(d,ε)$, 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 $1+ε$. Moreover, for every $n\geq n_0$, 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.

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