A stability theorem on cube tessellations

Abstract

It is shown that if a $d$-dimensional cube is decomposed into n cubes, the side lengths of which belong to the interval $\left(1-\frac{1}{n^{1/d}+1},1\right],then$n$isaperfect$d$-th power and all cubes are of the same size. This result is essentially tight.

4 pages