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We prove quantitative equidistribution results for actions of Abeliansubgroups of the $2g+1$ dimensional Heisenberg group acting on compact$2g+1$-dimensional homogeneous nilmanifolds. The results are based on the studyof the $C^\infty$-cohomology of the action of such groups, on tame estimates ofthe associated cohomological equations and on a renormalisation methodinitially applied by Forni to surface flows and by Forni and the second authorto other parabolic flows. As an application we obtain bounds for finite Thetasums defined by real quadratic forms in $g$ variables, generalizing theclassical results of Hardy and Littlewood \cite{MR1555099, MR1555214} and theoptimal result of Fiedler, Jurkat and K\"orner \cite{MR0563894} to higherdimension.

Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds