Recurrence for pretentious systems along generalized Pythagorean triples

TL;DR

This paper proves recurrence results for measure-preserving systems along solutions to generalized Pythagorean equations, confirming a conjecture about partition regularity in a structured ergodic setting.

Contribution

It establishes the ergodic-theoretic form of the generalized Pythagorean partition regularity conjecture for pretentious multiplicative actions.

Findings

Confirmed the ergodic form of the conjecture for structured actions.

Any finite coloring from pretentious multiplicative functions contains a monochromatic Pythagorean triple.

Extended recurrence results to solutions of $ax^2 + b y^2 = c z^2$.

Abstract

We establish multiple recurrence results for pretentious measure-preserving multiplicative actions along generalized Pythagorean triples, that is, solutions to the equation $ax^2 + b y^2 = c z^2$. This confirms the ergodic-theoretic form of the generalized Pythagorean partition regularity conjecture in this critical case of structured measure-preserving actions. As a consequence of our main theorem, any finite coloring of $\mathbb{N}$ generated by the level sets of finitely many pretentious completely multiplicative functions, must contain a monochromatic generalized Pythagorean triple.