PreprintCC BY-NC-ND 4.0arXiv
Moments of the Cross-Sectional Siegel-Veech Transform
Abstract
We study the Siegel--Veech transform on the Poincaré section for the horocycle flow consisting of lattice surfaces with a visible horizontal holonomy vector of length at most one. We compute its first moment and derive formulas for higher and factorial moments with respect to the section's induced ergodic invariant probability measure, which is not supported on a periodic orbit. We emphasize these formulas in the case of the square torus of unit area and, as an application, use its first-moment formula to recover the Boca--Zaharescu pair-correlation density for Farey fractions.
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BibTeX
@misc{artiles2026moments,
title = {{Moments of the Cross-Sectional Siegel-Veech Transform}},
author = {Albert Artiles},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33788},
eprint = {2609.33788},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33788},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Moments of the Cross-Sectional Siegel-Veech Transform
AU - Albert Artiles
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33788
UR - https://arxiv.org/abs/2609.33788
AB - We study the Siegel--Veech transform on the Poincaré section for the horocycle flow consisting of lattice surfaces with a visible horizontal holonomy vector of length at most one. We compute its first moment and derive formulas for higher and factorial moments with respect to the section's induced ergodic invariant probability measure, which is not supported on a periodic orbit. We emphasize these formulas in the case of the square torus of unit area and, as an application, use its first-moment formula to recover the Boca--Zaharescu pair-correlation density for Farey fractions.
ER -