Quadratic discrepancy estimates for probability measures on the Heisenberg group

TL;DR

The paper extends Roth-type $L^{2}$ discrepancy theory to the Heisenberg group $\mathbb{H}^{n}$ with respect to upper Ahlfors regular measures, introducing an $L^{2}$-discrepancy for a natural family of cylindrically symmetric test sets and proving a lower bound of order $N^{1-1/(Q-2)}$ with $Q=2n+2$. The authors develop a harmonic-analysis framework based on the group Fourier transform via the Schrödinger representation, and they decompose the main discrepancy into two core contributions, analyzed through a Laguerre–Hermite spectral calculus for radial functions and a heat-kernel–based kernel $K_{s}$. The main contributions are a Roth-type lower bound and an upper bound of order $N^{1-1/Q}$ for the minimal quadratic discrepancy, highlighting both the potential sharpness gap and the intrinsic geometry of $\mathbb{H}^{n}$. This work lays foundational steps toward a discrepancy theory on the Heisenberg group, linking sub-Riemannian harmonic analysis with irregular sampling in a non-commutative setting.

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.

Theorems & Definitions (44)

• Theorem 1
• Theorem 2
• Proposition 3
• Definition 4
• Remark 5
• Theorem 6
• Theorem 7
• Definition 8
• Proposition 9
• Theorem 10