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Equidistribution for higher-rank abelian actions on Heisenberg nilmanifolds. (English) Zbl 1353.37055

Summary: 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 renormalization 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 G. H. Hardy and J. E. Littlewood [Acta Math. 37, 155–191, 193–239 (1914; JFM 45.0305.03); ibid. 47, 189–198 (1925; JFM 51.0159.02)] and the optimal result of H. Fiedler et al. [Acta Arith. 32, 129–146 (1977; Zbl 0308.10021)] to higher dimension.

MSC:

37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37A17 Homogeneous flows
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11K36 Well-distributed sequences and other variations
22E25 Nilpotent and solvable Lie groups
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