Uniformity of multiplicative functions and partition regularity of some quadratic equations

Since the theorems of Schur and van der Waerden, numerous partition regularity results have been proved for linear equations, but progress has been scarce for non-linear ones, the hardest case being equations in three variables. We prove partition regularity for certain equations involving forms in three variables, showing for example that the equations $16x^2+9y^2=n^2$ and $x^2+y^2-xy=n^2$ are partition regular, where $n$ is allowed to vary freely in $\mathbb{N}$. For each such problem we establish a density analogue that can be formulated in ergodic terms as a recurrence property for actions by dilations on a probability space. Our key tool for establishing such recurrence properties is a decomposition result for multiplicative functions which is of independent interest. Roughly speaking, it states that the arbitrary multiplicative function of modulus 1 can be decomposed into two terms, one that is approximately periodic and another that has small Gowers uniformity norm of degree three.