Sparse Equidistribution of Unipotent Orbits in Finite-Volume Quotients of $\text{PSL}(2,\mathbb R)$, with appendices

TL;DR

This paper proves that under certain Diophantine conditions, unipotent orbits in finite-volume quotients of PSL(2,R) are equidistributed for small perturbations, extending previous results and analyzing non-Diophantine points.

Contribution

It generalizes Venkatesh's equidistribution result to orbits with polynomial time scaling and includes Hausdorff dimension computations for non-Diophantine points.

Findings

Proves equidistribution of orbits for small gamma under Diophantine conditions.

Computes Hausdorff dimensions of non-Diophantine point sets.

Establishes effective equidistribution results using exponential mixing.

Abstract

In this note, we consider the orbits $\{pu(n^{1+\gamma})|n\in\mathbb N\}$ in $\Gamma\backslash\text{PSL}(2,\mathbb R)$, where $\Gamma$ is a non-uniform lattice in $\text{PSL}(2,\mathbb R)$ and $u(t)$ is the standard unipotent group in $\text{PSL}(2,\mathbb R)$. Under a Diophantine condition on the intial point $p$, we can prove that $\{pu(n^{1+\gamma})|n\in\mathbb N\}$ is equidistributed in $\Gamma\backslash\text{PSL}(2,\mathbb R)$ for small $\gamma>0$, which generalizes a result of Venkatesh (Ann.of Math. 2010). We will compute Hausdorff dimensions of subsets of non-Diophantine points in Appendix A, using results of lattice counting problem. In Appendix B we will use a technique of Venkatesh (Ann.of Math. 2010) and an exponential mixing property to prove a weak version of a result of Str\"ombergsson (J Mod Dynam, 2013), which is about the effective equidistribution of horospherical orbits.