Sizes of Pre-Images of the Minimal Euclidean Function on the Gaussian Integers

TL;DR

This paper provides formulas for the size of pre-images of the minimal Euclidean function on Gaussian integers, advancing understanding of Euclidean functions in algebraic number fields.

Contribution

It introduces explicit formulas for the sizes of pre-images of the minimal Euclidean function on Gaussian integers, building on previous geometric descriptions.

Findings

Formulas for the size of $ ext{phi}_{ ext{Z}[i]}^{-1}([0,n])$ are established.

The work extends the computability of Euclidean functions to Gaussian integers.

Provides a geometric approach to understanding pre-image sizes.

Abstract

In 2023, the author presented the first computable minimal Euclidean function for a non-trivial number field. Along with a formula for $\phi_{\mathbb{Z}[i]}$, the minimal Euclidean function on the Gaussian inteers, the same paper introduced a geometric description for $\phi_{\mathbb{Z}[i]}^{-1}([0,n])$. This paper uses that construction to prove formulas for the size of the function's pre-images, or $|\phi_{\mathbb{Z}[i]}^{-1}([0,n])|$.