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Partition regularity in imaginary quadratic rings of integers

Sebastián Donoso, Andreu Ferré Moragues, Andreas Koutsogiannis, Wenbo Sun

Abstract

We obtain partition regularity results for homogeneous quadratic equations whose parametrized solutions admit nice factorizations into linear forms over rings of integers of imaginary quadratic fields. To do so, we develop number-theoretic results of independent interest on such fields, as Halász's theorem (which actually holds on arbitrary number fields), a characterization for aperiodic completely multiplicative functions, the Turán-Kubilius inequality, and a new concentration estimate for multiplicative functions.

Partition regularity in imaginary quadratic rings of integers

Abstract

We obtain partition regularity results for homogeneous quadratic equations whose parametrized solutions admit nice factorizations into linear forms over rings of integers of imaginary quadratic fields. To do so, we develop number-theoretic results of independent interest on such fields, as Halász's theorem (which actually holds on arbitrary number fields), a characterization for aperiodic completely multiplicative functions, the Turán-Kubilius inequality, and a new concentration estimate for multiplicative functions.
Paper Structure (37 sections, 52 theorems, 318 equations)

This paper contains 37 sections, 52 theorems, 318 equations.

Table of Contents

  1. Introduction
  2. Partition and density regularity
  3. Number-Theoretic tools
  4. Characterizations for completely multiplicative functions
  5. Halász's theorem over number fields
  6. The Turán-Kubilius inequality
  7. Organization of the paper
  8. Precise statements of the main results
  9. The main density regularity result and applications
  10. Characterizations for completely multiplicative functions
  11. Halász's theorem over number fields
  12. The Turán-Kubilius inequality
  13. Notation
  14. An outline for the proof of the density regularity result
  15. Reductions
  16. ...and 22 more sections

Key Result

Theorem 1.2

Let $d\in{\mathbb N}$ be squarefree. For $a,b,c\in{\mathbb Z}\backslash\{0\}$, if $\sqrt{-ac}, \sqrt{-bc}\in {\mathbb Z}[\tau_{d}]$, then the equation $ax^{2}+by^{2}+cz^{2}=0$ is partition regular over ${\mathbb Z}[\tau_{d}]$ in $(x,y)$.

Theorems & Definitions (104)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3: Density regularity
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2
  • proof : Proof of Theorems \ref{['th1']} and \ref{['th1density']} assuming Theorem \ref{['thmain']}
  • Corollary 2.3
  • Remark
  • Theorem 2.4
  • ...and 94 more