Distribution of Shapes of orthogonal Lattices

TL;DR

This paper advances the understanding of how primitive vectors on large spheres and their associated lattice shapes distribute in space, removing previous restrictions and providing effective error bounds for dimensions four and five.

Contribution

It extends previous equidistribution results to all cases for d=4,5 by removing congruence restrictions using unipotent flow techniques, and establishes polynomial error terms.

Findings

Equidistribution holds for d=4,5 without congruence restrictions.

The distribution has a polynomial error term.

Results apply to primitive vectors on large spheres in these dimensions.

Abstract

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.