We prove that if the Hausdorff dimension of a compact set [math] is greater than [math] , then the set of three-point configurations determined by [math] 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 [math] if [math] , and the index [math] 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 ) = ∬ f ( x − u ) g ( x − v ) d K ( u , v ) , ¶ where [math] is surface measure on the set [math] , and we prove a scale of estimates that includes [math] on positive functions. ¶ As an application of our main result, it follows that for finite sets of cardinality [math] and belonging to a natural class of discrete sets in the plane, the maximum number of times a given three-point configuration arises is [math] (up to congruence), improving upon the known bound of [math] in this context.