Distribution of shapes of orthogonal lattices

It was recently shown by Aka, Einsiedler and Shapira that if $d>2$ , the set of primitive vectors on large spheres when projected to the $(d-1)$ -dimensional sphere coupled with the shape of the lattice in their orthogonal complement equidistribute in the product space of the sphere with the space of shapes of $(d-1)$ -dimensional lattices. Specifically, for $d=3,4,5$ some congruence conditions are assumed. By using recent advances in the theory of unipotent flows, we effectivize the dynamical proof to remove those conditions for $d=4,5$ . It also follows that equidistribution takes place with a polynomial error term with respect to the length of the primitive points.

Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner’s measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite-dimensional vector spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions.

Non-divergence of unipotent flows on quotients of rank-one semisimple groups

Let$G$be a semisimple Lie group of rank one and$\unicode[STIX]{x1D6E4}$be a torsion-free discrete subgroup of$G$. We show that in$G/\unicode[STIX]{x1D6E4}$, given$\unicode[STIX]{x1D716}>0$, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than$\unicode[STIX]{x1D6FF}$for a$1-\unicode[STIX]{x1D716}$proportion of the time, for some$\unicode[STIX]{x1D6FF}>0$. The result also holds for any finitely generated discrete subgroup$\unicode[STIX]{x1D6E4}$and this generalizes Dani’s quantitative non-divergence theorem [On orbits of unipotent flows on homogeneous spaces.Ergod. Th. & Dynam. Sys.4(1) (1984), 25–34] for lattices of rank-one semisimple groups. Furthermore, for a fixed$\unicode[STIX]{x1D716}>0$, there exists an injectivity radius$\unicode[STIX]{x1D6FF}$such that, for any unipotent trajectory$\{u_{t}g\unicode[STIX]{x1D6E4}\}_{t\in [0,T]}$, either it spends at least a$1-\unicode[STIX]{x1D716}$proportion of the time in the set with injectivity radius larger than$\unicode[STIX]{x1D6FF}$, for all large$T>0$, or there exists a$\{u_{t}\}_{t\in \mathbb{R}}$-normalized abelian subgroup$L$of$G$which intersects$g\unicode[STIX]{x1D6E4}g^{-1}$in a small covolume lattice. We also extend these results to when$G$is the product of rank-one semisimple groups and$\unicode[STIX]{x1D6E4}$a discrete subgroup of$G$whose projection onto each non-trivial factor is torsion free.