On sets containing a unit distance in every direction

TL;DR

This paper studies the minimal box dimensions of compact sets in the plane that contain a unit distance in every direction, revealing bounds that highlight the geometric complexity of such sets.

Contribution

It establishes new lower and upper bounds on the box dimensions of sets containing a unit distance in all directions, advancing understanding of their geometric properties.

Findings

Lower box dimension is at least 4/7.

Upper box dimension can be as low as 2/3.

Sets may have zero Hausdorff dimension.

Abstract

We investigate the box dimensions of compact sets in $\mathbb{R}^2$ that contain a unit distance in every direction (such sets may have zero Hausdorff dimension). Among other results, we show that the lower box dimension must be at least $\frac{4}{7}$ and can be as low as $\frac{2}{3}$. This quantifies in a certain sense how far the unit circle is from being a difference set.