On volumes of simplices in intermediate dimensions

Abstract

A variant of the Falconer distance problem asks for fixed $k\ge 1$ and $d\ge k+1$, how large does the Hausdorff dimension of a Borel set $E\subset {\mathbb{R}}^d$ need to be to guarantee that there exist $x_0,\dots ,x_k\in E$ such that ${\text{Vol}}_{k+1}^{(x_0,\dots ,x_k)}(E)=\{{\text{Vol}}_{k+1}(x_0,\dots ,x_k,x_{k+1}):x_{k+1}\in E\}$ has positive Lebesgue measure. Here ${\text{Vol}}_{k+1}(x_0,\dots ,x_k,x_{k+1})$ denotes the $k+1$-volume of the $k+1$ simplex formed by $x_0,\dots ,x_k,x_{k+1}$. Recently, Shmerkin and Yavicoli established a sharp dimensional threshold $k$ in the case when $d=k+1$. In this paper we extend their result to $k+1\le d\le 2k$ and obtain a non-trivial dimensional threshold $d-k$ when $d>2k$. The result is motivated by ideas from Shmerkin and Yavicoli. A crucial part of the argument is an application of work by Bright, Ortiz and Zakharov on a continuum Beck-type theorem for hyperplanes as well as classic results of Marstrand on projections and slicing theorems. In addition, we investigate a more elementary approach under a condition called the Fubini property for Hausdorff dimension as introduced in the work of Héra, Keleti and Máthé.

arXiv comment16 pages, 5 figures. Typos corrected and exposition expanded a bit