PreprintarXiv licencearXiv
Densities of arithmetic Hecke triangle group orbits
Abstract
We give new proofs computing the asymptotic densities for orbits of the arithmetic Hecke triangle groups Γq when q=4 and q=6. We use elementary number theory techniques along with basic properties of the Möbius function and Riemann zeta function with additional congruence conditions. This note actually came about by observing that, in the q=4 case, the underlying congruence condition partitions the set of coprime integer pairs into three classes of equal density.
Related
- Arithmetic Siegel-Weil for the spherical Hecke algebra
- Hecke Orbits on Globally Drinfeld-Uniformized Shimura Varieties Are p-adically Nowhere Dense
- Arithmetic Wu Formulas and the Generalized Hecke Theorem
- On the type of a complex hyperbolic triangle group
- Isogeny rigidity under algebraic group actions and finite translation orbits
BibTeX
@misc{fairchild2026densities,
title = {{Densities of arithmetic Hecke triangle group orbits}},
author = {Samantha Fairchild and Christopher R. H. Hanusa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.22142},
eprint = {2607.22142},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.22142},
}
RIS
TY - UNPB
TI - Densities of arithmetic Hecke triangle group orbits
AU - Samantha Fairchild
AU - Christopher R. H. Hanusa
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2607.22142
UR - https://arxiv.org/abs/2607.22142
AB - We give new proofs computing the asymptotic densities for orbits of the arithmetic Hecke triangle groups Γ_q when q=4 and q=6. We use elementary number theory techniques along with basic properties of the Möbius function and Riemann zeta function with additional congruence conditions. This note actually came about by observing that, in the q=4 case, the underlying congruence condition partitions the set of coprime integer pairs into three classes of equal density.
ER -