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

Pablo Shmerkin, Alexia Yavicoli

TL;DR

The paper advances Falconer-type questions for higher-order configurations by showing that, for all $1\le k<d$, a fractal set $E\subset\mathbb{R}^d$ with $\dim_H(E)>k$ yields a pinned $(k{+}1)$-volume set $\mathrm{Vol}_{k+1}^{(x_1,\dots,x_k)}(E)$ of positive Lebesgue measure, with nonempty interior when $k\ge2$. For the critical case $k=d{-}1$, it provides a refined level-set description giving positive-measure volumes with substantial lower bounds on the dimension of the corresponding level sets, using a measure-theoretic construction based on sliced measures and a fiber-dimension analysis guided by Marstrand–Mattila theory. A finer slicing theorem is developed via gauge-function (dimension-function) tools, enabling positive-measure results under more delicate size assumptions and extending the framework to sets with dimension functions just beyond $k$. The work also derives partial results in the critical dimension and a planar result on the dimension of the set of areas, employing radial projection and projection theorems, highlighting new gauge-function slicing methods that may have independent applicability to related Falconer-type problems.

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 $k$''.

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

TL;DR

The paper advances Falconer-type questions for higher-order configurations by showing that, for all , a fractal set with yields a pinned -volume set of positive Lebesgue measure, with nonempty interior when . For the critical case , it provides a refined level-set description giving positive-measure volumes with substantial lower bounds on the dimension of the corresponding level sets, using a measure-theoretic construction based on sliced measures and a fiber-dimension analysis guided by Marstrand–Mattila theory. A finer slicing theorem is developed via gauge-function (dimension-function) tools, enabling positive-measure results under more delicate size assumptions and extending the framework to sets with dimension functions just beyond . The work also derives partial results in the critical dimension and a planar result on the dimension of the set of areas, employing radial projection and projection theorems, highlighting new gauge-function slicing methods that may have independent applicability to related Falconer-type problems.

Abstract

We prove that for , if is a Borel subset of of Hausdorff dimension strictly larger than , the set of -volumes determined by points in has positive one-dimensional Lebesgue measure. In the case , we obtain an essentially sharp lower bound on the dimension of the set of tuples in 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 ''.
Paper Structure (9 sections, 14 theorems, 54 equations)

This paper contains 9 sections, 14 theorems, 54 equations.

Table of Contents

  1. Introduction and main results
  2. Sharp dimension thresholds: proofs of Theorems \ref{['Thm:Main']} and \ref{['Thm:MainRefined']}
  3. Preliminaries
  4. Proof of Theorem \ref{['Thm:Main']}
  5. Proof of Theorem \ref{['Thm:MainRefined']}
  6. A finer slicing theorem
  7. Partial results in the critical dimension
  8. A sufficient condition in terms of gauge functions
  9. Dimension of the set of areas

Key Result

Theorem 1.1

Fix $d\in {\mathbb N}_{\ge 2}$ and $k\in\{1,\ldots,d-1\}$. Let $E \subseteq \mathbb{R}^d$ be a Borel set with $\mathop{\mathrm{dim_H}}\nolimits(E) >k \geq 1$. Then there exist $x_{1},\ldots,x_{k}\in E$ such that the set has positive Lebesgue measure. Moreover, when $k\ge 2$, there exist $x_{1},\ldots,x_{k}\in E$ such that $\mathop{\mathrm{Vol}}\nolimits_{k+1}^{(x_1,\ldots,x_{k})}(E)$ has nonempty

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • proof : Proof of Theorem \ref{['Thm:Main']}
  • proof : Proof of Theorem \ref{['Thm:MainRefined']}
  • Definition 3.1: Gauge functions
  • Definition 3.2: Generalized Hausdorff measures
  • ...and 15 more