**Title:** Higher order Fourier analysis of multiplicative functions and applications

**Speaker:** Bernard Host

**Speaker Info:** University of Paris Est

**Brief Description:**

**Special Note**:

**Abstract:**

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 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 Katai. We use variants of this structure theorem to derive applications that have number theoretic and combinatorial flavor:1. We give simple necessary and sufficient conditions for the Gowers norms over N of a bounded multiplicative function to be zero.

2. We generalize a classical result of Daboussi and Delange and prove asymptotic orthogonality of multiplicative functions to ``irrational'' nilsequences.

3. We prove that for certain polynomials in two variables all ``aperiodic'' multiplicative functions satisfy Chowla's zero mean conjecture.

4. We give the first partition regularity results for homogeneous quadratic equations in three variables, showing for example that for every partition of the integers into finitely many cells there exist distinct x,y belonging to the same cell and some positive integer n such that 16x^2+9y^2=n^2.

This is joint work with Nikos Frantzikinakis.

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