On three point configurations determined by subsets of the Euclidean plane, the associated bilinear operator and applications to discrete geometry

Abstract

We prove that if the Hausdorff dimension of a compact set $E\subset {\mathbb{R}}^2$ is greater than 7/4, then the set of {\ag three-point configurations determined by $E$ has positive three-dimensional measure}. We establish this by showing that {\ag a} natural measure on the set of {\ag such configurations} has {\ag Radon-Nikodym derivative} in $L^{\infty }$ if $\dH(E)> 7/4$, and the index 7/4 in this last result cannot, in general, be improved. This problem naturally leads to the study of a bilinear convolution operator, $$B(f,g)(x)=\int \int f(x-u)g(x-v)dK(u,v),$$ where $K$ is surface measure on the set $\{(u,v)\in {\mathbb{R}}^2\times {\mathbb{R}}^2:|u|=|v|=|u-v|=1\}$, and we prove a scale of estimates that includes $B:L_{-1/2}^2({\mathbb{R}}^2)\times L^2({\mathbb{R}}^2)\to L^1({\mathbb{R}}^2)$ on positive functions. As an application of our main result, it follows that {\ag for finite sets of cardinality $n$ and belonging to a natural class of discrete sets in the plane}, the maximum number of times a given three-point configuration arises is $O(n^{9/7+\epsilon })$ (up to congruence), improving upon the known bound of $O(n^{4/3})$ in this context.

References in corpus (2)

• On the Erdos distinct distance problem in the plane
• Theory of dimension for large discrete sets and applications

Cited by in corpus (1)

• Dimensional lower bounds for Falconer type incidence and point configuration theorems