  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.
