Lattice deformations in the Heisenberg group

TL;DR

This paper studies the deformation space of the integer Heisenberg group under automorphisms, analyzing its structure as a dynamical system and deriving statistical estimates for lattice point counting, including a Minkowski-type theorem.

Contribution

It provides a detailed analysis of the deformation space as a measurable dynamical system and establishes variance bounds for lattice point counting in the Heisenberg setting.

Findings

Mean and variance estimates for lattice point counts

A Minkowski-type theorem for the Heisenberg group

Geometric conditions for variance bounds in non-Euclidean sets

Abstract

The space of deformations of the integer Heisenberg group under the action of $\textrm{Aut}(H(\mathbb{R}))$ 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 $\mathbb{R}^3$; 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.