Higher order Fourier analysis of multiplicative functions and appliactions
Résumé
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 K\'atai.
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 $\N$) 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 $\lambda\in \N$ such that $16x^2+9y^2=\lambda^2$ and the same holds for the equation $x^2-xy+y^2=\lambda^2$.