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arXiv · 2310.17855

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

Pablo Shmerkin·Alexia Yavicoli
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Abstract

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 $E$ has positive one-dimensional Lebesgue measure. In the case $k=d-1$, we obtain an essentially sharp lower bound on the dimension of the set of tuples in $E$ generating a given volume. We also establish a finer version of the classical slicing theorem of Marstrand-Mattila in terms of dimension functions, and use it to extend our results to sets of ``dimension logarithmically larger than $k$''.

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Pablo Shmerkin, Alexia Yavicoli. 2023-10-27. On the volumes of simplices determined by a subset of $\mathbb{R}^d$. https://arxiv.org/abs/2310.17855

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