A stability theorem on cube tessellations

TL;DR

This paper proves a stability theorem for cube tessellations, showing that under certain size constraints, a decomposition of a cube into smaller cubes implies all are equal and the number of cubes is a perfect power.

Contribution

It establishes a new stability result linking the number and sizes of cubes in a tessellation, with tight bounds and conditions.

Findings

If a cube is decomposed into n smaller cubes with side lengths in a specific interval, then n is a perfect d-th power.

All smaller cubes are of the same size under these conditions.

The result is tight, meaning the bounds cannot be significantly improved.

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 $n$ is a perfect $d$-th power and all cubes are of the same size. This result is essentially tight.