On Fourier asymptotics and effective equidistribution

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 $μ$, satisfying $\sum_{\mathbb{Z}\ni|m|\leq X}|\widehatμ(m)| = O\left(X^{1/2-θ}\right)$ with $θ>7/64,$ our result holds. This class of measures contains convolutions of $s$-Ahlfors regular measures for $s>39/64$, as well as a subclass of self-similar measures. Moreover, our result is sharp upon the Ramanujan--Petersson Conjecture (upon which the above $θ$ can be chosen arbitrarily small): there are measures $μ$ with $\widehatμ(ξ) = O\left(|ξ|^{-1/2+ε}\right)$ for which equidistribution fails.