Distributing Points on the Torus via Modular Inverses

We study various statistics regarding the distribution of the points $$\{(\frac{d}{q},\frac{\bar{d}}{q})\in {\mathbb{T}}^2:d\in (\mathbb{Z}\mathrm{/}q\mathbb{Z}{)}^{\times }\}$$ 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.