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

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

2010-09-13

arxiv.org/abs/1009.2471v4

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

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