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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.15850 |
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| _version_ | 1866909997845184512 |
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| author | Brandolini, Luca Monguzzi, Alessandro Monti, Matteo |
| author_facet | Brandolini, Luca Monguzzi, Alessandro Monti, Matteo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15850 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quadratic discrepancy estimates for probability measures on the Heisenberg group Brandolini, Luca Monguzzi, Alessandro Monti, Matteo Classical Analysis and ODEs Functional Analysis 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. |
| title | Quadratic discrepancy estimates for probability measures on the Heisenberg group |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2601.15850 |