Cube packings in Euclidean spaces

TL;DR

This paper investigates the geometric properties of compact sets in Euclidean spaces that contain cube boundaries with side lengths in (0,1), establishing lower bounds on their box dimension and size.

Contribution

It introduces new lower bounds on the box dimension of sets containing cube boundaries and provides sharp examples demonstrating these bounds.

Findings

Sets must have lower box dimension at least n-0.5

Such sets are large in a specific, precisely defined sense

The paper provides sharp examples matching the bounds

Abstract

In this paper we study some cube packing problems. In particular we are interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain boundaries of cubes with all side lengths in $(0,1)$. We show here that such sets must have lower box dimension at least $n-0.5$ and we will also provide sharp examples. We also show here that such sets must be large in general in a precise sense which is also introduced in this paper.