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Journal d'Analyse Mathématique

, Volume 134, Issue 1, pp 55–105 | Cite as

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

  • Wenbo Sun
Article
  • 24 Downloads

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 U3-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 x2y2 = n2 for some Gaussian integer n. The analog of this statement over Z remains open.

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Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA

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