Complex continued fractions, Kleinian and extremal theory for cusp excursions

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 Fréchet distribution. We use this to find a limiting distribution for cusp excursions on Bianchi orbifolds associated with the aforementioned rings of quadratic integers.

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