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Higher order Fourier analysis of multiplicative functions and applications

Authors: Nikos Frantzikinakis and Bernard Host
Journal: J. Amer. Math. Soc. 30 (2017), 67-157
MSC (2010): Primary 11N37; Secondary 05D10, 11N60, 11B30, 37A45.
Published electronically: March 1, 2016
MathSciNet review: 3556289
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Abstract: 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 $x^2-xy+y^2=\lambda ^2$.

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Additional Information

Nikos Frantzikinakis
Affiliation: Department of Mathematics, University of Crete, Voutes University Campus, Heraklion 71003, Greece
MR Author ID: 712393
ORCID: 0000-0001-7392-5387

Bernard Host
Affiliation: Laboratoire d’analyse et de mathématiques appliquées, Université Paris-Est Marne-la-Vallée, UMR CNRS 8050, 5 Bd Descartes, 77454 Marne la Vallée Cedex, France
MR Author ID: 88605

Keywords: Multiplicative functions, Gowers uniformity, partition regularity, inverse theorems, Chowla conjecture.
Received by editor(s): April 21, 2014
Received by editor(s) in revised form: December 30, 2015, and January 25, 2016
Published electronically: March 1, 2016
Additional Notes: The first author was partially supported by Marie Curie IRG 248008.
The second author was partially supported by Centro de Modelamiento Matemático, Universitad de Chile.
Article copyright: © Copyright 2016 American Mathematical Society