# Diophantine approximation on lines in \mathbb{C}^2 with Gaussian prime constraints - enhanced version

## TL;DR

This paper investigates how well points on lines in complex two-dimensional space can be approximated by Gaussian primes, extending classical Diophantine approximation results to Gaussian prime constraints.

## Contribution

It introduces analogs of classical prime approximation results for Gaussian primes in the complex plane, focusing on Diophantine approximation on lines in ^2.

## Key findings

- Established new bounds for fractional parts involving Gaussian primes
- Extended classical Diophantine approximation results to Gaussian prime setting
- Provided foundational results for approximation in complex Gaussian integer context

## Abstract

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes. To this end, we develop analogs of well-known results on small fractional parts of $p\gamma$, $p$ running over the primes and $\gamma$ being a fixed irrational, for Gaussian primes.