Squares and their centers

TL;DR

This paper explores how the size of sets containing square boundaries or vertices relates to the set of centers, revealing that different size notions lead to varying results, with some bounds proven to be sharp.

Contribution

It provides new bounds and constructions for the dimensions of sets containing square boundaries or vertices with centers in a given set, highlighting differences across size notions.

Findings

A compact set of Hausdorff dimension 1 can contain squares with centers in [0,1]^2.

Such a set must have packing and lower box dimension at least 7/4.

The bounds for packing and box dimensions are shown to be sharp.

Abstract

We study the relationship between the sizes of two sets $B, S\subset\mathbb{R}^2$ when $B$ contains either the whole boundary, or the four vertices, of a square with axes-parallel sides and center in every point of $S$, where size refers to one of cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprinsingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set $B$ of Hausdorff dimension $1$ which contains the boundary of an axes-parallel square with center in every point $[0,1]^2$, but prove that such a $B$ must have packing and lower box dimension at least $\tfrac{7}{4}$, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others.