Decomposition of a cube into nearly equal smaller cubes

TL;DR

This paper proves that large enough d-dimensional cubes can be partitioned into nearly equal smaller cubes with controlled size ratios, using a bounded number of different side lengths, advancing understanding of cube decompositions.

Contribution

It introduces a method to decompose large cubes into nearly equal smaller cubes with a bounded number of side lengths, improving previous results on cube partitioning.

Findings

Decomposition into n smaller cubes with side length ratio at most 1+ε.

Use of at most d+2 different side lengths for the decomposition.

For large n, such decompositions are always possible.

Abstract

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