#### Vol. 5, No. 2, 2012

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On triangles determined by subsets of the Euclidean plane, the associated bilinear operators and applications to discrete geometry

### Allan Greenleaf and Alex Iosevich

Vol. 5 (2012), No. 2, 397–409
##### Abstract

We prove that if the Hausdorff dimension of a compact set $E\subset {ℝ}^{2}$ is greater than $\frac{7}{4}$, then the set of three-point configurations determined by $E$ has positive three-dimensional measure. We establish this by showing that a natural measure on the set of such configurations has Radon–Nikodym derivative in ${L}^{\infty }$ if ${dim}_{\mathsc{ℋ}}\left(E\right)>\frac{7}{4}$, and the index $\frac{7}{4}$ in this last result cannot, in general, be improved. This problem naturally leads to the study of a bilinear convolution operator,

$B\left(f,g\right)\left(x\right)=\iint f\left(x-u\right)\phantom{\rule{0.3em}{0ex}}g\left(x-v\right)\phantom{\rule{0.3em}{0ex}}dK\left(u,v\right),$

where $K$ is surface measure on the set $\left\{\left(u,v\right)\in {ℝ}^{2}×{ℝ}^{2}:|u|=|v|=|u-v|=1\right\}$, and we prove a scale of estimates that includes $B:{L}_{-1∕2}^{2}\left({ℝ}^{2}\right)×{L}^{2}\left({ℝ}^{2}\right)\to {L}^{1}\left({ℝ}^{2}\right)$ on positive functions.

As an application of our main result, it follows that 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 $O\left({n}^{\frac{9}{7}+ϵ}\right)$ (up to congruence), improving upon the known bound of $O\left({n}^{\frac{4}{3}}\right)$ in this context.

##### Keywords
Falconer–Erdős distance problem, distance set, geometric combinatorics, multilinear operators, triangles
##### Mathematical Subject Classification 2010
Primary: 42B15, 52C10