Uniform bounds for period integrals and sparse equidistribution

TL;DR

This paper establishes uniform bounds for twisted averages of smooth functions along long horocycle segments on compact hyperbolic surfaces, leading to new equidistribution results for sparse horocycle subsets.

Contribution

It introduces spectral methods to derive uniform bounds independent of spectral gaps, advancing understanding of horocycle dynamics and sparse equidistribution.

Findings

Uniform bounds for twisted horocycle averages

Equidistribution results for sparse horocycle subsets

Spectral methods effective regardless of spectral gap

Abstract

Let $M=\Gamma\backslash\mathrm{PSL}(2,\mathbb{R})$ be a compact manifold, and let $f\in C^\infty(M)$ be a function of zero average. We use spectral methods to get uniform (i.e. independent of spectral gap) bounds for twisted averages of $f$ along long horocycle orbit segments. We apply this to obtain an equidistribution result for sparse subsets of horocycles on $M$.