Quadratic discrepancy estimates for probability measures on the Heisenberg group

TL;DR

This paper develops the initial framework for quadratic discrepancy estimates of finite point sets on the Heisenberg group, extending classical Euclidean results to a non-commutative setting.

Contribution

It introduces an $L^2$-discrepancy measure for the Heisenberg group and establishes a Roth-type lower bound, pioneering discrepancy theory in this non-commutative context.

Findings

Established a Roth-type lower bound depending on the homogeneous dimension

Extended classical discrepancy estimates to the Heisenberg group

Introduced a new $L^2$-discrepancy framework for non-commutative groups

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.