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 $E \subset {\Bbb R}^2$ 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\R^2 \times \R^2: |u|=|v|=|u-v|=1\}$, and we prove a scale of estimates that includes $B:L^2_{-1/2}({\Bbb R}^2) \times L^2({\Bbb R}^2) \to L^1({\Bbb R}^2)$ 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.