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PreprintCC BY-NC-ND 4.0arXiv

Moments of the Cross-Sectional Siegel-Veech Transfrom

  • Albert Artiles
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Abstract

We study the Siegel--Veech transform on the Poincaé 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. 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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Cite

BibTeXRIS
BibTeX
@misc{artiles2026moments,
  title = {{Moments of the Cross-Sectional Siegel-Veech Transfrom}},
  author = {Albert Artiles},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.36463},
  eprint = {2609.36463},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36463},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Moments of the Cross-Sectional Siegel-Veech Transfrom
AU  - Albert Artiles
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.36463
UR  - https://arxiv.org/abs/2609.36463
AB  - We study the Siegel--Veech transform on the Poincaé 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. 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  -