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arXiv · 2508.09778

Recurrence for pretentious systems along generalized Pythagorean triples

Nikos Frantzikinakis·Andreas Mountakis
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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.

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Nikos Frantzikinakis, Andreas Mountakis. 2025-08-13. Recurrence for pretentious systems along generalized Pythagorean triples. https://arxiv.org/abs/2508.09778

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