Equidistribution of primitive lattices in ℝn

Abstract We count primitive lattices of rank d inside $\mathbb{Z}^{n}$ as their covolume tends to infinity, with respect certain parameters such lattices. These include, for example, the subspace that a lattice spans, namely its projection Grassmannian; homothety class and equivalence modulo rescaling rotation, often referred shape. add prior work Schmidt by allowing sets in spaces are general enough conclude joint equidistribution these parameters. In addition d-lattices Λ themselves, we also consider orthogonal complements $\mathbb{Z}^{n}$, ${\Lambda}^{\perp}$, show occurs jointly ${\Lambda}^{\perp}$. Finally, our asymptotic formulas number include an explicit bound on error term.