Monochromatic Almost-Pythagorean Quadruples

We prove that for any set $A$ with positive multiplicative upper density, there exist $x$, $y$, $z$ and $w$ in $A$ such that $x^2+y^2{w}^2=z^2$, 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.