Linnik's problem in fiber bundles over quadratic homogeneous varieties

TL;DR

This paper extends Linnik's problem to fiber bundles over quadratic homogeneous varieties, analyzing the distribution of integral matrices and lattices on level sets, with implications for number theory and homogeneous dynamics.

Contribution

It generalizes Linnik's equidistribution results to new geometric settings involving quadratic varieties and fiber bundles, providing explicit distribution computations.

Findings

Computed the statistics of $SL_{d}(bZ)$ matrices on polynomial level sets.

Generalized distribution results for primitive vectors and orthogonal lattices.

Extended Linnik's theorem to fiber bundles over quadratic homogeneous varieties.

Abstract

We compute the statistics of $SL_{d}(\mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(\mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of radially projected integral vectors from a large sphere into the unit sphere. Using the above result we generalize the work of Aka, Einsiedler and Shapira in various directions. For example, we compute the joint distribution of the residue classes modulo $q$ and the properly normalized orthogonal lattices of primitive integral vectors lying on the level set $-(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})+x_{4}^{2}=N$ as $N\to\infty$, where the normalized orthogonal lattices sit in a submanifold of the moduli space of rank-$3$ discrete subgroups of $\mathbb{R}^{4}$.