The space of deformations of the integer Heisenberg group under the action of Aut is a homogeneous space for a non-reductive group. We analyze its structure as a measurable dynamical system and obtain mean and variance estimates for Heisenberg lattice point counting in measurable subsets of ; in particular, we obtain a random Minkowski-type theorem. Unlike the Euclidean case, we show there are necessary geometric conditions on the sets that satisfy effective variance bounds.
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Jayadev S. Athreya, Ioannis Konstantoulas, Lattice deformations in the Heisenberg group. Groups Geom. Dyn. 14 (2020), no. 3, pp. 1007–1022DOI 10.4171/GGD/572