On angles determined by fractal subsets of the Euclidean space via Sobolev bounds for bilinear 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 bilinear 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2011
 DOI:
 10.48550/arXiv.1110.6792
 arXiv:
 arXiv:1110.6792
 Bibcode:
 2011arXiv1110.6792I
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Combinatorics;
 28A75;
 42B20;
 52C10