On volumes of simplices in intermediate dimensions

TL;DR

This paper investigates the Hausdorff dimension thresholds needed for sets in Euclidean space to contain simplices with positive volume measure, extending previous results to higher dimensions using geometric measure theory techniques.

Contribution

It extends the known dimensional thresholds for the Falconer distance problem variant to dimensions where d ≥ k+1, specifically for d up to 2k, and introduces new methods involving hyperplane theorems.

Findings

Established a non-trivial dimensional threshold d-k for d > 2k.

Extended sharp thresholds from the case d=k+1 to higher dimensions d between k+1 and 2k.

Applied continuum Beck-type theorems and classical projection results in the analysis.

Abstract

A variant of the Falconer distance problem asks for fixed $k\geq 1$ and $d\geq k+1$, how large does the Hausdorff dimension of a Borel set $E\subset\mathbb{R}^d$ need to be to guarantee that there exist $x_0,\ldots,x_{k}\in E$ such that $\text{Vol}_{k+1}^{(x_0,\ldots,x_{k})}(E) = \lbrace \text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1}) : x_{k+1}\in E \rbrace$ has positive Lebesgue measure. Here $\text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1})$ denotes the $k+1$-volume of the $k+1$ simplex formed by $x_0,\ldots,x_{k},x_{k+1}$. 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 $k+1 \leq d \leq 2k$ 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 H\'{e}ra, Keleti and M\'{a}th\'{e}.