  We study various statistics regarding the distribution of the points
\[\left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d
\in (\mathbb{Z}/q\mathbb{Z})^{\times}\right\}\] as $q$ tends to infinity. Due
to nontrivial bounds for Kloosterman sums, it is known that these points
equidistribute on the torus. We prove refinements of this result, including
bounds for the discrepancy, small scale equidistribution, bounds for the
covering exponent associated to these points, sparse equidistribution, and
mixing.
