A Complex Borel-Bernstein Theorem

TL;DR

This paper establishes a complex analogue of the Borel-Bernstein theorem for complex Hurwitz continued fractions, extending classical zero-one laws in metric Diophantine approximation to the complex setting.

Contribution

It introduces a new complex Borel-Bernstein theorem and derives a complex version of Khinchin's theorem, advancing the understanding of Diophantine approximation in complex numbers.

Findings

Proves a complex Borel-Bernstein theorem for Hurwitz continued fractions

Derives a complex Khinchin's theorem as a corollary

Extends classical zero-one laws to complex Diophantine approximation

Abstract

Zero-one laws are a central topic in metric Diophantine approximation. A classical example of such laws is the Borel-Bernstein theorem. In this note, we prove a complex analogue of the Borel-Bernstein theorem for complex Hurwitz continued fractions. As a corollary, we obtain a complex version of Khinchin's theorem on Diophantine approximation.