An improved dimensional threshold for the angle problem

Abstract

The Falconer distinct distance problem asks for a compact set $E\subset\mathbb{R}^d$ how large its Hausdorff dimension needs to be to ensure that the Lebesgue measure of its distance set is positive. In this paper we consider the analogous question for the set of angles. We show that if the Hausdorff dimension of $E$ is strictly bigger than $\frac{d}{2}$ then the Lebesgue measure of the angles set is positive. In the plane this result was previously established by Harangi et al. In higher dimensions, our exponent improves the $\frac{d+1}{2}$ threshold previously obtain by the authors of this paper and Mihalis Mourgoglou. We do not know what the right dimensional threshold should be in higher dimensions.