Partition regularity of generalized Pythagorean pairs

TL;DR

This paper investigates the partition regularity of generalized Pythagorean equations, proving that under certain conditions, solutions with monochromatic variables exist in any finite coloring of positive integers, extending previous work with new multiplicative function techniques.

Contribution

The authors establish new results on the partition regularity of quadratic equations, extending prior work by incorporating uniformity properties of multiplicative functions and addressing conditional cases based on an Elliott-type conjecture.

Findings

Existence of monochromatic solutions for $ax^2+by^2=cz^2$ under natural coefficient conditions

Conditional results depending on an Elliott-type conjecture for certain coefficient choices

Development of new uniformity and concentration estimates for multiplicative functions

Abstract

We address partition regularity problems for homogeneous quadratic equations. A consequence of our main results is that, under natural conditions on the coefficients $a,b,c$, for any finite coloring of the positive integers, there exists a solution to $ax^2+by^2=cz^2$ where $x$ and $y$ have the same color (and similar results for $x,z$ and $y,z$). For certain choices of $(a,b,c)$, our result is conditional on an Elliott-type conjecture. Our proofs build on and extend previous arguments of the authors dealing with the Pythagorean equation. We make use of new uniformity properties of aperiodic multiplicative functions and concentration estimates for multiplicative functions along arbitrary binary quadratic forms.