2014/03/04 by Nikos Frantzikinakis, Frantzikinakis, Nikos, Bernard Host +1 · 4 citations
Computer Science · Mathematics · #05D10 #11B30 #11N37 #11N60 #37A45 #Analytic Number Theory Research #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1403.0945
openalex publication_date 2014/03/04 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28
We prove a structure theorem for multiplicative functions which states that\nan arbitrary bounded multiplicative function can be decomposed into two terms,\none that is approximately periodic and another that has small Gowers uniformity\nnorm of an arbitrary degree. The proof uses tools from higher order Fourier\nanalysis and some soft number theoretic input that comes in the form of an\northogonality criterion of K 'atai. We use variants of this structure theorem\nto derive applications of number theoretic and combinatorial flavor: (i) we\ngive simple necessary and sufficient conditions for the Gowers norms (over\n\ℕ) of a bounded multiplicative function to be zero, (ii)\ngeneralizing a classical result of Daboussi and Delange we prove asymptotic\northogonality of multiplicative functions to "irrational" nilsequences, (iii)\nwe prove that for certain polynomials in two variables all "aperiodic"\nmultiplicative functions satisfy Chowla's zero mean conjecture, (iv) we give\nthe first partition regularity results for homogeneous quadratic equations in\nthree variables showing for example that on every partition of the integers\ninto finitely many cells there exist distinct x,y belonging to the same cell\nand \λ\∈ \ℕ such that 16x2+9y2=\λ2 and the same holds\nfor the equation x2-xy+y2=\λ2.\n