Equidistribution of primitive lattices in $\mathbb{R}^n$

Tal Horesh, Yakov Karasik

Published 2020-12-08Version 1

We count primitive lattices of rank d inside $\mathbb{Z}^n$, as their covolume tends to infinity, w.r.t. certain parameters of such lattices. These parameters include, for example, the direction of a lattice, which is the subsapce that it spans; 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 that are general enough in the spaces of parameters to conclude joint equidistribution of these parameters. The main novelty is that, in addition to the primitive d-lattices {\Lambda} themselves, we also consider their orthogonal lattices $\Lambda^\perp$ and the quotients $\mathbb{Z}^n/\Lambda$. We are able to count not only w.r.t. parameters of the primitive lattices {\Lambda}, but also w.r.t. the same parameters of $\Lambda^\perp$ and $\mathbb{Z}^n/\Lambda$. By doing so, we achieve joint equidistribution of, e.g., the shapes of primitive lattices, and the shapes of their orthogonal lattices. Finally, our asymptotic formulas for the number of primitive lattices includes an explicit error term.