On sets containing a unit distance in every direction

Abstract

We investigate the box dimensions of compact sets in $\mathbb{R}^n$ 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 $\frac{n^2(n-1)}{2n^2-1}$ and can be at most $\frac{n(n-1)}{2n-1}$. This quantifies in a certain sense how far the unit sphere $S^{n-1}$ is from being a difference set.