On Fourier asymptotics and effective equidistribution

We prove effective equidistribution of expanding horocycles in ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})\mathrm{\backslash }{\mathrm{S}\mathrm{L}}_2(\mathbb{R})$ 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 ${\sum }_{\mathbb{Z}\ni \mathrm{\mid }m\mathrm{\mid }\le X}\mathrm{\mid }\widehat{\mu }(m)\mathrm{\mid }=O\left(X^{1\mathrm{/}2-\theta }\right)$ with $\theta >7\mathrm{/}64,$ our result holds. This class of measures contains convolutions of $s$-Ahlfors regular measures for $s>39\mathrm{/}64$, 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 $\widehat{\mu }(\xi )=O\left(\mathrm{\mid }\xi {\mathrm{\mid }}^{-1\mathrm{/}2+\epsilon }\right)$ for which equidistribution fails.