Higher order Fourier analysis of multiplicative functions and applications

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 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 λ ∈ N such that 16x 2 + 9y 2 = λ 2 and the same holds for the equation x 2 − xy + y 2 = λ 2 .

The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures

“…By summation by parts it suffices to obtain a decomposition obeying (15). By repeating the proof of [32, Corollary 4.6] verbatim 9 , we can write f 1 = f 1,1 + f 1,2 where f 1,1 is a nilsequence of some finite degree D, and f 1,2 obeys the asymptotic lim x→∞ E p≤x | f 1,2 (ap)| = 0 for any non-zero integer a. We can now neglect the f 1,2 term as it can be absorbed into the g error.…”

“…, J D , some irrational nilcharacters χ i, j of degree i, and some linear functionals c i, j . Using [32,Lemma 5.8] (noting that if χ is an irrational nilcharacter, then so is χ(a·)) we see that each of the 9 In [32, Corollary 4.6], a was required to be a natural number rather than a non-zero integer, however one can easily adapt the arguments to the case of negative a with only minor modifications (in particular, one has to modify the definition of X m slightly to allow l to be negative).…”

The Logarithmically Averaged Chowla and Elliott Conjectures for Two-Point Correlations

“…Using the literature from higher order Fourier analysis (in particular the inverse theorem in [15], together with transference arguments from [9,14], or [33]), one is now faced with the task of controlling sums even more complicated than (51), in which the linear phases n → e(αn) are now replaced by more general nilsequences of higher step (which one then has to take the supremum over, before performing the integral); this task can be viewed as a local version of the machinery in [7,8], and will be carried out in detail in [31]. Of course, since satisfactory control on (51) is not yet available (even if one inserts logarithmic averaging), it is not feasible at present to control higher step analogues of (51) either.…”

Sarnak’s Conjecture: What’s New

“…Frantzikinakis and Host in [67] prove a deep structure theorem for multiplicative functions from M. One of the consequences of it is the following characterization of aperiodic functions: u P M is aperiodic if and only if it is uniform, that is, all Gowers uniformity seminorms 17 vanish [67]. In [23] (see Theorem 1.3 therein), this result is extended to show that u P M conv is either uniform or rational.…”