Partition regularity of generalized Pythagorean pairs
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929451082711040 |
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| author | Frantzikinakis, Nikos Klurman, Oleksiy Moreira, Joel |
| author_facet | Frantzikinakis, Nikos Klurman, Oleksiy Moreira, Joel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08360 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partition regularity of generalized Pythagorean pairs Frantzikinakis, Nikos Klurman, Oleksiy Moreira, Joel Combinatorics Number Theory Primary: 05D10, Secondary:11N37, 11B30, 37A44 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. |
| title | Partition regularity of generalized Pythagorean pairs |
| topic | Combinatorics Number Theory Primary: 05D10, Secondary:11N37, 11B30, 37A44 |
| url | https://arxiv.org/abs/2407.08360 |