Higher order Fourier analysis of multiplicative functions and applications

By Nikos Frantzikinakis, B. Host. 2014

87 citations recorded by the source corpus.

We prove a structure theorem for multiplicative functions which states that an arbitrary bounded multiplicative function 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 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 $\mathbb{N}$) of a bounded multiplicative function to be zero, $(ii)$ generalizing a classical result of Daboussi and Delange 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 $x^2-xy+y^2=\lambda^2$.

Subfields: Algebra and Number Theory, Artificial Intelligence, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/b6f5952ab856ad1dcc8dc454f2314545a53a9a1a

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Citations in the corpus

A transference approach to a Roth-type theorem in the squares (2015, background): “Similarly, it is unknown if the equation x(2) + y(2) = 2z(2) is partition regular (see [FrHo14]).”
Sarnak’s Conjecture: What’s New (2017, background): “One of the consequences of it is the following characterization of aperiodic functions: u P M is aperiodic if and only if it is uniform, that is, all Gowers uniformity seminorms(17) vanish [67].”
The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures (2017, background): “the structural theory of multiplicative functions in [13]).”