There are infinitely many obstructions to the existence of smooth solutions of the cohomological equation U u = f , where U is the vector field generating the horocycle flow on the unit tangent bundle S M of a Riemann surface M of finite area and f is a given function on S M. We study the Sobolev regularity of these obstructions, construct smooth solutions of the cohomological equation, and derive asymptotics for the ergodic averages of horocycle flows.
We prove quantitative equidistribution results for nilflows on compact three-dimensional homogeneous nilmanifolds by a method based on renormalization and invariant distributions for nilflows. As an application we obtain a dynamical proof of quantitative equidistribution results for the sequence P(n) (mod 1), where $P(X)\in \mathbb{R}[X]$ is a quadratic polynomial with generic leading coefficient. Bounds on theta sums, that is, Birkhoff sums of the exponential function along such sequences, were proved by Hardy and Littlewood and in optimal form by Fiedler, Jurkat and Körner.
We prove quantitative equidistribution results for actions of Abelian subgroups of the 2g + 1 dimensional Heisenberg group acting on compact 2g + 1-dimensional homogeneous nilmanifolds. The results are based on the study of the C ∞ -cohomology of the action of such groups, on tame estimates of the associated cohomological equations and on a renormalisation method initially applied by Forni to surface flows and by Forni and the second author to other parabolic flows. As an application we obtain bounds for finite Theta sums defined by real quadratic forms in g variables, generalizing the classical results of Hardy and Littlewood [HL14, HL26] and the optimal result of Fiedler, Jurkat and Körner [FJK77] to higher dimension.
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