On the volumes of simplices determined by a subset of ${\mathbb{R}}^d$

Pablo Shmerkin, Alexia Yavicoli

We prove that for $1\le k<d$, if $E$ is a Borel subset of ${\mathbb{R}}^d$ of Hausdorff dimension strictly larger than $k$, the set of $(k+1)$-volumes determined by $k+2$ points in $…