Cubes and Their Centers

TL;DR

This paper investigates the dimensional relationships between sets containing cube skeleta in Euclidean space, providing sharp bounds and extending previous planar results to higher dimensions with applications of hypergraph theory.

Contribution

It generalizes planar results to higher dimensions, establishing sharp dimension bounds for sets containing cube skeleta and introducing new combinatorial bounds using hypergraph theory.

Findings

Sharp estimates for packing and box-counting dimensions of sets

Cardinality bounds for sets with discrete cube skeleta

Partial results on Hausdorff dimension and dual polytope analysis

Abstract

We study the relationship between the sizes of sets $B,S$ in $\mathbb{R}^n$ where $B$ contains the $k$-skeleton of an axes-parallel cube around each point in $S$, generalizing the results of Keleti, Nagy, and Shmerkin about such sets in the plane. We find sharp estimates for the possible packing and box-counting dimensions of $B$ and $S$. These estimates follow from related cardinality bounds for sets containing the discrete skeleta of cubes around a finite set of a given size. The Katona-Kruskal theorem from hypergraph theory plays an important role. We also find partial results for the Hausdorff dimension and settle an analogous question for the dual polytope of the cube, the orthoplex.