On Fourier asymptotics and effective equidistribution

TL;DR

This paper establishes effective equidistribution of expanding horocycles in a modular surface for measures with specific Fourier decay, using new automorphic and harmonic analysis techniques, and discusses the sharpness of these results.

Contribution

It introduces novel methods combining automorphic forms and harmonic analysis to prove effective equidistribution for measures with certain Fourier asymptotics.

Findings

Effective equidistribution holds for measures with Fourier decay rate O(X^{1/2-θ}) with θ>7/64.

Includes measures like convolutions of s-Ahlfors regular measures for s>39/64.

Results are sharp assuming the Ramanujan--Petersson Conjecture.

Abstract

We prove effective equidistribution of expanding horocycles in $\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{R})$ with respect to various classes of Borel probability measures on $\mathbb{R}$ having certain Fourier asymptotics. Our proof involves new techniques combining tools from automorphic forms and harmonic analysis. In particular, for any Borel probability measure $\mu$, satisfying $\sum_{\mathbb{Z}\ni|m|\leq X}|\widehat{\mu}(m)| = O\left(X^{1/2-\theta}\right)$ with $\theta>7/64,$ our result holds. This class of measures contains convolutions of $s$-Ahlfors regular measures for $s>39/64$, and as well as, a sub-class of self-similar measures. Moreover, our result is sharp upon the Ramanujan--Petersson Conjecture (upon which the above $\theta$ can be chosen arbitrarily small): there are measures $\mu$ with $\widehat\mu(\xi) = O\left(|\xi|^{-1/2+\epsilon}\right)$ for which equidistribution fails.