We count primitive lattices of rank d inside 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 Λ themselves, we also consider their orthogonal lattices Λ ⊥ and the quotients Z n /Λ. We are able to count not only w.r.t. parameters of the primitive lattices Λ, but also w.r.t. the same parameters of Λ ⊥ and Z n /Λ. 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.
We establish effective counting and equidistribution results for lattice points in families of domains in hyperbolic spaces, of any dimension and over any field. The domains we focus on are defined as product sets with respect to the Iwasawa decomposition. Several classical Diophantine problems can be reduced to counting lattice points in such domains, including distribution of shortest solution to the gcd equation, and angular distribution of primitive vectors in the plane. We give an explicit and effective solution to these problems, and extend them to imaginary quadratic number fields. Further applications include counting lifts of closed horospheres to hyperbolic manifolds and establishing an equidistribution property of integral solutions to the Diophantine equation defined by a Lorentz form.
Linnik type problems concern the distribution of projections of integral points on the unit sphere as their norm increases, and different generalizations of this phenomenon. Our work addresses a question of this type: we prove the uniform distribution of the projections of primitive Z 2 points in the p-adic unit sphere, as their (real) norm tends to infinity. The proof is via counting lattice points in semi-simple S-arithmetic groups.Résumé. -(Directions p-adique de vecteurs primitifs) Les problèmes de type Linnik concernent la distribution des projections des points entiers sur la sphère unitaire lorsque leur norme augmente et différentes généralisations de ce phénomène. Notre travail s'intéresse à une question de ce type : nous prouvons la distribution uniforme des projections des points primitifs de Z 2 sur la sphère unitaire p-adique lorsque leur norme (réelle) tend vers l'infini. La preuve se fait en comptant les points d'un réseau dans des S-groupes arithmétiques semi-simples.
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.
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