Horocycle flow at product of two primes

TL;DR

This paper proves that horocycle orbits in certain arithmetic quotients of SL(2,R) equidistribute when sampled at integers with exactly two prime factors, extending understanding of orbit distribution in homogeneous dynamics.

Contribution

It establishes equidistribution of horocycle orbits at integers with two prime factors for co-compact arithmetic lattices and SL(2,Z), a novel result in homogeneous dynamics.

Findings

Horocycle orbits equidistribute at integers with two prime factors.

Results apply to co-compact arithmetic lattices and SL(2,Z).

Advances understanding of orbit distribution in number-theoretic sampling.

Abstract

We show that if $\Gamma$ is a co-compact arithmetic lattice in $SL(2,\mathbb{R})$ or $\Gamma=SL(2,\mathbb{Z})$ then the horocycle orbit of every non-periodic point $x\in SL(2,\mathbb{R})/\Gamma$ equidistributes (with respect to Haar measure) when sampled at integers having exactly two prime factors.