Abstract: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.
From: Nikos Frantzikinakis [view email]
[v1]
Thu, 11 Jul 2024 10:12:34 UTC (61 KB)
[v2]
Wed, 7 Aug 2024 03:11:57 UTC (62 KB)
[v3]
Mon, 27 Jul 2026 16:09:59 UTC (62 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。