Complex continued fractions, Kleinian and extremal theory for cusp excursions
Dynamical Systems
2025-10-14 v2 Number Theory
Abstract
For the each of the five Euclidean rings of complex quadratic integers, we consider a complex continued fraction algorithm with digits in the ring. We show for each algorithm that the maximal digit obeys a Fr\'echet distribution. We use this to find a limiting distribution for cusp excursions on Bianchi orbifolds associated with the aforementioned rings of quadratic integers.
Keywords
Cite
@article{arxiv.2401.00626,
title = {Complex continued fractions, Kleinian and extremal theory for cusp excursions},
author = {Alexander Baumgartner and Mark Pollicott},
journal= {arXiv preprint arXiv:2401.00626},
year = {2025}
}
Comments
19 pages, 4 figures Fixed typos. Converted appendix to new section and improved presentation of proofs. Added Theorem 1.3 (an application to Diophantine Approximation) along with a proof in Subsection 6.3. Changed title to more accurately reflect contents of preprint