Effective equidistribution of horocycle lifts

TL;DR

This paper provides an effective rate of equidistribution for lifts of horocycles from the classical modular surface to a larger space, extending Ratner's theorem with explicit convergence rates using advanced number theory tools.

Contribution

It introduces an effective version of equidistribution for horocycle lifts, generalizing previous methods to linear and quadratic lifts with explicit convergence rates.

Findings

Established explicit rates of equidistribution for horocycle lifts.

Extended Ratner's measure classification theorem to a broader setting.

Applied Weil's resolution of the Riemann hypothesis for function fields.

Abstract

We give a rate of equidistribution of lifts of horocycles from the space $\mathrm{SL}(2,\mathbb Z)\backslash \mathrm{SL}(2, \mathbb R)$ to the space $\mathrm{ASL}(2,\mathbb Z)\backslash \mathrm{ASL}(2,\mathbb R)$, making effective a theorem of Elkies and McMullen. This result constitutes an effective version of Ratner's measure classification theorem for measures supported on general horocycle lifts. The method used relies on Weil's resolution of the Riemann hypothesis for function fields in one variable and generalizes the approach of Str\"ombergsson to the case of linear lifts and that of Browning and the author to rational quadratic lifts.