Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

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

Request Permissions   Purchase Content 
 

 

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
DOI: https://doi.org/10.1090/jams/857
Published electronically: March 1, 2016
MathSciNet review: 3556289
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 11N37, 05D10, 11N60, 11B30, 37A45

Retrieve articles in all journals with MSC (2010): 11N37, 05D10, 11N60, 11B30, 37A45


Additional Information

Nikos Frantzikinakis
Affiliation: Department of Mathematics, University of Crete, Voutes University Campus, Heraklion 71003, Greece
Email: frantzikinakis@gmail.com

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
Email: bernard.host@univ-mlv.fr

DOI: https://doi.org/10.1090/jams/857
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