Normal distance distribution in Heisenberg groups

Establish that the distance distribution in the higher-dimensional modular Heisenberg group H_{2d+1}(Z/mZ) is approximately normal.

Background

For finite abelian groups, distances associated with natural generators have asymptotically normal distributions because they are sums of independent discrete uniform random variables. Numerical evidence suggests an analogous phenomenon for higher-dimensional modular Heisenberg groups.

References

The distribution of distances in the higher-dimensional modular Heisenberg group $H_{2d+1}(Z/mZ)$ is approximately normal.

— CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)  (2509.19162 - Chervov et al., 23 Sep 2025) in Section 10, subsection “Distribution of distances in finite nilpotent groups”