A structure theorem for multiplicative functions over the Gaussian integers and applications

TL;DR

This paper establishes a structure theorem for multiplicative functions over Gaussian integers, decomposing them into periodic and uniform parts, and applies it to prove partition regularity results for quadratic equations in Gaussian integers.

Contribution

It introduces a novel structure theorem for multiplicative functions over Gaussian integers and applies it to solve partition regularity problems involving quadratic forms.

Findings

Decomposition of multiplicative functions into structured and uniform parts

Proof of partition regularity for specific quadratic equations over Gaussian integers

Extension of classical number theory results to Gaussian integers

Abstract

We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U^{3}-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadratic forms in three variables. For example, we show that for any finite coloring of the Gaussian integers, there exist distinct nonzero elements x and y of the same color such that x^{2}-y^{2}=n^{2} for some Gaussian integer n. The analog of this statement over Z remains open.