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

Greenleaf, Allan; Iosevich, Alex

doi:10.2140/apde.2012.5.397

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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

DOI: 10.2140/apde.2012.5.397

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 ${dim}_{ℋ} (E) > \frac{7}{4}$ , 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 (f, g) (x) = \iint f (x - u) g (x - v) d K (u, v),

where $K$ is surface measure on the set ${(u, v) \in ℝ^{2} \times ℝ^{2} : | u | = | v | = | u - v | = 1}$ , and we prove a scale of estimates that includes $B : L_{- 1 ∕ 2}^{2} (ℝ^{2}) \times L^{2} (ℝ^{2}) \to L^{1} (ℝ^{2})$ 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 $O (n^{\frac{9}{7} + ϵ})$ (up to congruence), improving upon the known bound of $O (n^{\frac{4}{3}})$ in this context.

Keywords

Falconer–Erdős distance problem, distance set, geometric combinatorics, multilinear operators, triangles

Mathematical Subject Classification 2010

Primary: 42B15, 52C10

Milestones

Received: 15 September 2010

Revised: 1 July 2011

Accepted: 9 October 2011

Published: 27 August 2012

Authors

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

Vol. 5, No. 2, 2012