On Fourier asymptotics and effective equidistribution

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.