Abstract
We prove that if the Hausdorff dimension of a compact set is greater than , then the set of three-point configurations determined by 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 if , and the index in this last result cannot, in general, be improved. This problem naturally leads to the study of a bilinear convolution operator,
where is surface measure on the set , and we prove a scale of estimates that includes on positive functions.
As an application of our main result, it follows that for finite sets of cardinality and belonging to a natural class of discrete sets in the plane, the maximum number of times a given three-point configuration arises is (up to congruence), improving upon the known bound of in this context.
Citation
Allan Greenleaf. Alex Iosevich. "On triangles determined by subsets of the Euclidean plane, the associated bilinear operators and applications to discrete geometry." Anal. PDE 5 (2) 397 - 409, 2012. https://doi.org/10.2140/apde.2012.5.397
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