Distributing Points on the Torus via Modular Inverses

Abstract

Abstract We study various statistics regarding the distribution of the points $$\begin{equation*} \left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d \in (\mathbb{Z}/q\mathbb{Z})^{\times}\right\} \end{equation*}$$ as q tends to infinity. Due to non-trivial 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 with these points, sparse equidistribution, and mixing.