Integer points and their orthogonal lattices

TL;DR

This paper extends classical equidistribution results of integer points on spheres to pairs involving orthogonal lattice shapes, using advanced dynamical systems techniques under certain congruence conditions.

Contribution

It introduces a conjecture on equidistribution involving orthogonal lattice shapes and proves it under additional congruence assumptions using higher rank diagonalizable actions.

Findings

Proposes a new conjecture on lattice shape equidistribution.

Proves the conjecture under specific congruence conditions.

Utilizes joining results for higher rank actions to achieve the proof.

Abstract

Linnik proved in the late 1950's the equidistribution of integer points on large spheres under a congruence condition. The congruence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec) using completely different techniques. We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition.