Diophantine approximation on lines in $\mathbb{C}^2$ with Gaussian prime constraints

TL;DR

This paper investigates how well points on lines in complex two-dimensional space can be approximated when numerators and denominators are Gaussian primes, extending Diophantine approximation theory into complex prime constraints.

Contribution

It introduces a novel approach to Diophantine approximation in complex space with Gaussian prime restrictions, a less explored area in number theory.

Findings

Established bounds for approximation quality under Gaussian prime constraints

Extended classical Diophantine approximation results to complex Gaussian primes

Provided new insights into the distribution of Gaussian primes on lines in ^2

Abstract

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes.