Equidistribution of primitive lattices in ℝn

Abstract

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