Summary: We prove that, if the Hausdorff dimension of a compact set \(E \subset \mathbb R^2\) is greater than \(\frac74\), 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 a Radon-Nikodym derivative in \(L^\infty\) if \(\dim_{\mathcal{H}}(E) > \frac74\), and the index \(\frac74\) in this last result cannot, in general, be improved. This problem naturally leads to the study of a bilinear convolution operator,
\[ B(f,g)(x) = \iint f(x - u)g(x - v)dK(u,v), \]
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 \(B : L_{-1/2}^2(\mathbb R^2) \times L^2(\mathbb R^2) \rightarrow L^1(\mathbb R^2)\) 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 \(O(n^{\frac97 +\varepsilon})\) (up to congruence), improving upon the known bound of \(O(n^{\frac43})\) in this context.