On angles determined by fractal subsets of the Euclidean space

Eyvindur Ari Palsson (Department of Mathematics, University of Rochester, New York, U.S.A.; and Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts, U.S.A.)

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. This result improves on the the threshold $\min \lbrace \frac{d}{2} + \frac{4}{3} , d - 1 \rbrace$ obtained by A. Máthé [15]. The result complements those of V. Harangi, T. Keleti, G. Kiss, P. Maga, A. Máthé, P. Mattila and B. Stenner in [8] and those of V. Harangi in [7]. We also obtain new upper bounds for the number of times an angle can occur among $N$ points in $\mathbb{R}^d , d \geq 4$, motivated by the results of Apfelbaum and Sharir [1] and Pach and Sharir [16]. We then use this result to establish sharpness thresholds in the continuous setting.

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Published 21 February 2017