PreprintCC BY 4.0arXiv
A local limit theorem for singularity visits on translation surfaces
Abstract
Let (X,ω) be a translation surface with at least one singularity. We investigate the distribution of the number of singularity encounters along prime closed geodesics on X. Writing ℓ(γ) for the geometric length of a closed geodesic and m(γ) for the number of singularities it encounters, we establish a uniform counting local limit theorem for the values of m(γ) sampled over ℓ(γ)<T. The key technical step is to introduce and study a trace-class operator whose spectral data encodes the Fourier transform of our counting distribution. Our results hold for every translation surface, without genericity assumptions.
Related
BibTeX
@misc{cantrell2026local,
title = {{A local limit theorem for singularity visits on translation surfaces}},
author = {Stephen Cantrell},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01060},
eprint = {2611.01060},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01060},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A local limit theorem for singularity visits on translation surfaces
AU - Stephen Cantrell
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01060
UR - https://arxiv.org/abs/2611.01060
AB - Let (X,ω) be a translation surface with at least one singularity. We investigate the distribution of the number of singularity encounters along prime closed geodesics on X. Writing ℓ(γ) for the geometric length of a closed geodesic and m(γ) for the number of singularities it encounters, we establish a uniform counting local limit theorem for the values of m(γ) sampled over ℓ(γ)<T. The key technical step is to introduce and study a trace-class operator whose spectral data encodes the Fourier transform of our counting distribution. Our results hold for every translation surface, without genericity assumptions.
ER -