Dimensional lower bounds for Falconer type incidence and point configuration theorems

TL;DR

This paper establishes new lower bounds on the Hausdorff dimension needed for sets in Euclidean space to contain rich configurations of points, such as simplices and triangles, extending Falconer type theorems using geometric and number-theoretic methods.

Contribution

It introduces novel dimensional thresholds for Falconer type theorems for k-simplices and incidence theorems, generalizing previous results with new number-theoretic techniques.

Findings

Dimensional lower bound of d(k+1)/(d+2) for k-simplices

Lower bound of (d+1)/2 for incidence theorems in certain dimensions

Extension of results to convex settings for triangles in higher dimensions

Abstract

Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. In this paper, we study the problem of how large the Hausdorff dimension of $E$ must be in order for the set of distinct noncongruent $k$-simplices in $E$ (that is, noncongruent point configurations of $k+1$ points from $E$) to have positive Lebesgue measure. This generalizes the $k=1$ case, the well-known Falconer distance problem and a major open problem in geometric measure theory. We establish a dimensional lower threshold of $\frac{d(k+1)}{d+2}$ for Falconer type theorems for $k$-simplices. This threshold is nontrivial in the range $d/2 \leq k \leq d$ and is obtained through counting simplices in a standard lattice using results of the Gauss circle problem. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We also establish a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for $k$-simplices where $k\leq d \leq 2k+1$ by generalizing an example of Mattila. Finally, we prove a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for triangles in a convex setting in every dimension greater than $3$. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.