Higher order Fourier analysis of multiplicative functions and applications

We prove a structure theorem for multiplicative functions which states that an arbitrary multiplicative function of modulus at most $1$ can be decomposed into two terms, one that is approximately periodic and another that has small Gowers uniformity norm of an arbitrary degree. The proof uses tools from higher order Fourier analysis and finitary ergodic theory, and some soft number theoretic input that comes in the form of an orthogonality criterion of Kátai. We use variants of this structure theorem to derive applications of number theoretic and combinatorial flavor: $(i)$ we give simple necessary and sufficient conditions for the Gowers norms (over $\mathbb {N}$) of a bounded multiplicative function to be zero, $(ii)$ generalizing a classical result of Daboussi we prove asymptotic orthogonality of multiplicative functions to “irrational” nilsequences, $(iii)$ we prove that for certain polynomials in two variables all “aperiodic” multiplicative functions satisfy Chowla’s zero mean conjecture, $(iv)$ we give the first partition regularity results for homogeneous quadratic equations in three variables, showing for example that on every partition of the integers into finitely many cells there exist distinct $x,y$ belonging to the same cell and $\lambda \in \mathbb {N}$ such that $16x^2+9y^2=\lambda ^2$, and the same holds for the equation $x^2-xy+y^2=\lambda ^2$.