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 $\hat{I}¼$, satisfying ${\sum }_{\mathbb{Z}\ni |m|\le X}|\hat{\hat{I}}¼(m)|=O\left(X^{1/2-\hat{I}¸}\right)$ with $\hat{I}¸>7/64,$ 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 $\hat{I}¸$ can be chosen arbitrarily small): there are measures $\hat{I}¼$ with $\hat{\hat{I}}¼(\hat{I}¾)=O\left(|\hat{I}¾{|}^{-1/2+\hat{I}\mu }\right)$ for which equidistribution fails.

30 pages, To appear in Crelle's Journal