Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds

TL;DR

This paper establishes quantitative equidistribution results for Abelian subgroup actions on Heisenberg nilmanifolds, utilizing cohomological methods and renormalization techniques, with applications to bounds on higher-dimensional Theta sums.

Contribution

It introduces new methods for analyzing equidistribution on higher-dimensional nilmanifolds and extends classical bounds on Theta sums to higher dimensions.

Findings

Quantitative equidistribution results for Abelian actions on Heisenberg nilmanifolds.

Bounds for higher-dimensional Theta sums generalizing classical results.

Application of cohomological and renormalization techniques to dynamical systems.

Abstract

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 $C^\infty$-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 \cite{MR1555099, MR1555214} and the optimal result of Fiedler, Jurkat and K\"orner \cite{MR0563894} to higher dimension.