Exact Generic Exponents of Approximation for the Action of the Integral Special Linear Group on the Torus

Given an integer $n\ge 2$ and points $\mathbf{x}, \mathbf{y}$ lying in the $n$--dimensional torus $\mathbb{T}^n$ equipped with its usual metric denoted by $\left\langle\;\cdot\;\right\rangle$, let \[ τ_n(\mathbf x,\mathbf y) = \sup\left\{ τ\ge0: \begin{array}{l} \text{there exist infinitely many } H