Recurrence for pretentious systems along generalized Pythagorean triples

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.