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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Higher order Fourier analysis of multiplicative functions and applications
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by Nikos Frantzikinakis and Bernard Host
J. Amer. Math. Soc. 30 (2017), 67-157
Published electronically: March 1, 2016


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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Bibliographic 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
  • Email:
  • 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
  • Email:
  • 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.
  • © Copyright 2016 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 30 (2017), 67-157
  • MSC (2010): Primary 11N37; Secondary 05D10, 11N60, 11B30, 37A45
  • DOI:
  • MathSciNet review: 3556289