Extreme Value Theory for Hurwitz Complex Continued Fractions

TL;DR

This paper investigates the extreme value behavior of Hurwitz complex continued fractions, establishing laws like Poisson and extreme value laws, and provides insights into their invariant measures and related nearest integer continued fractions.

Contribution

It introduces new extreme value laws for Hurwitz complex continued fractions and analyzes their invariant measures, extending understanding of their statistical properties.

Findings

Poisson law for Hurwitz continued fractions

Extreme value law established for the modulus of digits

Results also inform nearest integer continued fractions

Abstract

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we get several results concerning extremes of nearest integer continued fractions as well.