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

We prove that if the Hausdorff dimension of a compact set E 2 is greater than 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 if dim(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) =f(x u)g(x v)dK(u,v),

where K is surface measure on the set {(u,v) 2 × 2 : |u| = |v| = |u v| = 1}, and we prove a scale of estimates that includes B : L122(2) × L2(2) L1(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(n9 7 +ϵ) (up to congruence), improving upon the known bound of O(n4 3 ) in this context.

Falconer–Erdős distance problem, distance set, geometric combinatorics, multilinear operators, triangles
Mathematical Subject Classification 2010
Primary: 42B15, 52C10
Received: 15 September 2010
Revised: 1 July 2011
Accepted: 9 October 2011
Published: 27 August 2012
Allan Greenleaf
Department of Mathematics
University of Rochester
Rochester, NY 14627
United States
Alex Iosevich
Department of Mathematics
University of Rochester
Rochester, NY 14627
United States