Effective equidistribution of twisted horocycle flows and horocycle maps

TL;DR

This paper establishes effective bounds for twisted ergodic averages and equidistribution of horocycle flows and maps on hyperbolic surfaces, improving previous results and bounding Fourier coefficients of cusp forms.

Contribution

It introduces new bounds for twisted ergodic averages and horocycle map equidistribution, refining earlier results and applying Sobolev estimates and distribution scaling techniques.

Findings

Bounds for twisted ergodic averages in compact and non-compact cases

Effective equidistribution results for horocycle maps

Improved bounds on Fourier coefficients of cusp forms

Abstract

We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an application of our main theorems in the compact case we further improve on a result of A. Venkatesh, recently already improved by J. Tanis and P. Vishe, on a sparse equidistribution problem for classical horocycle flows proposed by N. Shah and G. Margulis, and in the general non-compact, finite area case we prove bounds on Fourier coefficients of cups forms which are off the best known bounds of A. Good only by a logarithmic term. Our approach is based on Sobolev estimates for solutions of the cohomological equation and on scaling of invariant distributions for twisted horocycle flows.