On angles determined by fractal subsets of the Euclidean space via Sobolev bounds for bi-linear operators

Abstract

We prove that if the Hausdorff dimension of a compact subset of ${\mathbb{R}}^d$ is greater than $\frac{d+1}{2}$, then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for bi-linear analogs of generalized Radon transforms and the method of stationary phase play a key role. These results complement those of V. Harangi, T. Keleti, G. Kiss, P. Maga, P. Mattila and B. Stenner in (\cite{HKKMMS10}). We also obtain new upper bounds for the number of times an angle can occur among $N$ points in ${\mathbb{R}}^d$, $d\ge 4$, motivated by the results of Apfelbaum and Sharir (\cite{AS05}) and Pach and Sharir (\cite{PS92}). We then use this result to establish sharpness results in the continuous setting. Another sharpness result relies on the distribution of lattice points on large spheres in higher dimensions.

References in corpus (1)

• Theory of dimension for large discrete sets and applications

Cited by in corpus (6)

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• Dimensional lower bounds for Falconer type incidence and point configuration theorems
• An improved dimensional threshold for the angle problem
• The Group Action Method and Radial Projection
• A Mattila-Sjölin theorem for triangles
• On radii of spheres determined by subsets of Euclidean space