# Higher order Fourier analysis of multiplicative functions and applications

@article{Frantzikinakis2014HigherOF,
title={Higher order Fourier analysis of multiplicative functions and applications},
author={Nikos Frantzikinakis and Bernard Host},
journal={arXiv: Number Theory},
year={2014}
}
• Published 4 March 2014
• Mathematics
• arXiv: Number Theory
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…
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## References

SHOWING 1-10 OF 70 REFERENCES
A structure theorem for multiplicative functions over the Gaussian integers and applications
It is proved that for any finite coloring of the Gaussian integers, there exist distinct nonzero elements x and y of the same color such that x2−y2 = n2 for some Gaussian integer n.
A Quantitative Ergodic Theory Proof of Szemerédi's Theorem
• T. Tao
• Mathematics
Electron. J. Comb.
• 2006
A quantitative, self-contained version of this ergodic theory proof is presented, which is elementary'' in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or theUse of the Fourier transform or inverse theorems from additive combinatorics.
Root numbers and the parity problem
Let E be a one-parameter family of elliptic curves over a number field. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk
On higher order Fourier analysis
We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms. We put
A sharp inequality of Halász type for the mean value of a multiplicative arithmetic function
Let g ( n ) be a complex valued multiplicative function such that | g ( n )| ≤ 1. In this paper we shall be concerned with the validity of the inequality under the weak condition g ( p )∈ for all
Decompositions, approximate structure, transference, and the Hahn–Banach theorem
We discuss three major classes of theorems in additive and extremal combinatorics: decomposition theorems, approximate structure theorems, and transference principles. We also show how the
Linear equations in primes
• Mathematics
• 2006
Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation
Uniformity seminorms on ℓ∞ and applications
• Mathematics
• 2009
A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on ℤ/Nℤ introduced by
An Arithmetic Regularity Lemma, An Associated Counting Lemma, and Applications
• Mathematics
• 2010
To Endre Szemeredi on the occasion of his 70th birthday Szemeredi’s regularity lemma can be viewed as a rough structure theorem for arbitrary dense graphs, decomposing such graphs into a structured
On solution‐free sets for simultaneous quadratic and linear equations
We consider a translation and dilation invariant system consisting of a diagonal quadratic equation and a linear equation with integer coefficients in s variables, where s ⩾ 9. We show via the