On the number of partitions of the hypercube ${\bf Z}_q^n$ into large subcubes

TL;DR

This paper determines the asymptotic number of ways to partition a high-dimensional hypercube into large subcubes, introducing new matrix operations and classifying fractal structures within this context.

Contribution

It provides an asymptotic formula for hypercube partitions and introduces the bang operation on star matrices, distinguishing fractals from other structures.

Findings

Number of partitions asymptotically equals n^{(q^m-1)/(q-1)}

Introduces the bang operation on star matrices

Classifies fractals as non-expandable under bang

Abstract

We prove that the number of partitions of the hypercube ${\bf Z}_q^n$ into $q^m$ subcubes of dimension $n-m$ each for fixed $q$, $m$ and growing $n$ is asymptotically equal to $n^{(q^m-1)/(q-1)}$. For the proof, we introduce the operation of the bang of a star matrix and demonstrate that any star matrix, except for a fractal, is expandable under some bang, whereas a fractal remains to be a fractal under any bang.