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arXiv · 2601.15850

Quadratic discrepancy estimates for probability measures on the Heisenberg group

Luca Brandolini·Alessandro Monguzzi·Matteo Monti
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Abstract

We initiate the study of quadratic discrepancy for finite point sets on the Heisenberg group $\mathbb H^n$ with respect to upper Ahlfors regular probability measures. For a natural family of test sets given by left translations and dilations of cylindrically defined neighborhoods, we introduce an $L^2$-discrepancy and establish a Roth-type lower bound depending on the homogeneous dimension of $\mathbb H^n$. This result extends classical discrepancy estimates from the Euclidean and compact settings to a non-commutative, step-two nilpotent Lie group. It should be viewed as a first step toward the development of a discrepancy theory on the Heisenberg group.

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Luca Brandolini, Alessandro Monguzzi, Matteo Monti. 2026-01-22. Quadratic discrepancy estimates for probability measures on the Heisenberg group. https://arxiv.org/abs/2601.15850

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