Abstract. We prove that the orbit of a generic point at prime values of the horocycle flow in the modular surface is dense in a set of positive measure; for some special orbits we also prove that they are dense in the whole space (assuming the Ramanujan/Selberg conjectures for GL 2 /Q). In the process, we derive an effective version of Dani's Theorem for the (discrete) horocycle flow.
We examine diverse local and global aspects of the family of Fourier series È n −α e(n k x). In particular, combining number theoretical and harmonic analytic arguments, we study differentiability, Hölder continuity, spectrum of singularities and fractal dimension of the graph.
We combine exponential sums, character sums and Fourier coefficients of automorphic forms to improve the best known upper bound for the lattice error term associated to rational ellipsoids.
Abstract. Let G = SL 2 (R) d and Γ = Γ d 0 with Γ 0 a lattice in SL 2 (R). Let S be any "curved" submanifold of small codimension of a maximal horospherical subgroup of G relative to an Rdiagonalizable element a in the diagonal of G. Then for S compact our result can be described by saying that a n vol S converges in an effective way to the volume measure of G/Γ when n → ∞, with vol S the volume measure on S.
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