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

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 $\mathrm{\mid }{\phi }_{\mathbb{Z}[i]}^{-1}([0,n])\mathrm{\mid }$.