Recurrence for pretentious systems along generalized Pythagorean triples
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909752139710464 |
|---|---|
| author | Frantzikinakis, Nikos Mountakis, Andreas |
| author_facet | Frantzikinakis, Nikos Mountakis, Andreas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09778 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recurrence for pretentious systems along generalized Pythagorean triples Frantzikinakis, Nikos Mountakis, Andreas Dynamical Systems Combinatorics Number Theory 37A44 (Primary) 05D10, 11N37 (Secondary) 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. |
| title | Recurrence for pretentious systems along generalized Pythagorean triples |
| topic | Dynamical Systems Combinatorics Number Theory 37A44 (Primary) 05D10, 11N37 (Secondary) |
| url | https://arxiv.org/abs/2508.09778 |