On Fourier asymptotics and effective equidistribution

We prove effective equidistribution of expanding horocycles in the space SL 2 ⁢ ( Z ) \ SL 2 ⁢ ( R ) SL₂(Z)₂(R) with respect to various classes of Borel probability measures on 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 ∑ Z ∋ | m | ≤ X | μ ̂ ⁢ ( m ) | = O ⁢ ( X 1 / 2 − θ ) see text ∑Z m≤ Xμ̂(m)=O(X1/2-θ) with θ > 7 / 64 θ>7/64 , our result holds. This class of measures contains convolutions of 𝑠-Ahlfors regular measures for s > 39 / 64 s>39/64 , and a sub-class 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 μ ̂ ⁢ ( ξ ) = O ⁢ ( | ξ | − 1 / 2 + ϵ ) </