Equidistribution of primitive rational points on expanding horospheres

TL;DR

This paper proves a conjecture about the uniform distribution of specific sparse rational points on expanding horospheres, revealing their arithmetic structure and implications for lattice shape distributions.

Contribution

It confirms Marklof's conjecture on equidistribution of sparse points on horospheres and connects this to lattice shape distribution results.

Findings

Sparse rational points on horospheres are equidistributed.

The points have an arithmetic structure.

Results extend to lattice shape distributions.

Abstract

We confirm a conjecture of Jens Marklof regarding the equidistribution of certain sparse collections of points on expanding horospheres. These collections are obtained by intersecting the expanded horosphere with a certain manifold of complementary dimension and turns out to be of arithmetic nature. This equidistribution result is then used along the lines suggested by Marklof to give an analogue of a result of W. Schmidt regarding the distribution of shapes of lattices orthogonal to integer vectors.