PreprintCC BY-NC-ND 4.0arXiv
Monochromatic Almost-Pythagorean Quadruples
Abstract
We prove that for any set A with positive multiplicative upper density, there exist x, y, z and w in A such that x2 + y2w2 = z2, generalizing a result of Frantzikinakis, Klurman and Moreira. Our techniques apply more broadly to a wide class of systems of equations, allowing us to solve these equations simultaneously and furthermore to restrict the w variable in an entirely different multiplicatively dense set. To do this, we prove a novel multiple recurrence result for multiplicative dynamical systems.
Related
BibTeX
@misc{leng2026monochromatic,
title = {{Monochromatic Almost-Pythagorean Quadruples}},
author = {James Leng and Redmond McNamara and Andreas Mountakis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04078},
eprint = {2610.04078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04078},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Monochromatic Almost-Pythagorean Quadruples
AU - James Leng
AU - Redmond McNamara
AU - Andreas Mountakis
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.04078
UR - https://arxiv.org/abs/2610.04078
AB - We prove that for any set A with positive multiplicative upper density, there exist x, y, z and w in A such that x^2 + y^2w^2 = z^2, generalizing a result of Frantzikinakis, Klurman and Moreira. Our techniques apply more broadly to a wide class of systems of equations, allowing us to solve these equations simultaneously and furthermore to restrict the w variable in an entirely different multiplicatively dense set. To do this, we prove a novel multiple recurrence result for multiplicative dynamical systems.
ER -