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.

We show that the invariant distributions for the horocycle flow on compact hyperbolic surfaces described by Flaminio and Forni [FF03] can be represented as distributions on the ideal circle tensorized with absolutely continuous measures, and use this information to derive their Hölder regularity.

We show that the Patterson–Sullivan measure on the limit set of a geometrically finite Kleinian group with cusps can be recovered as a weak limit of sums of Dirac masses placed on an appropriate orbit of each parabolic fixed point. A corollary is a sharp asymptotic estimate for a natural counting function associated to a cuspidal subgroup. We also discuss the connection between the above counting and the Riemann hypothesis in some examples of arithmetical lattices.

The integrability of the infinite relativistic Toda lattice is proved via the inverse scattering technique. The canonical structure of the scattering coordinates is derived in the framework of the r-matrix formalism.