A structural result for multiplicative functions and applications

I will discuss my recent work with B. Host, where we prove a structural result for multiplicative functions on the integers and give some applications to combinatorics and number theory. The structure theorem, roughly speaking, 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 combines tools from finitary ergodic theory, higher order Fourier analysis, equidistribution results on nilmanifolds, and some soft analytic number theory. Using this structural result we were able to give the first partition regularity results for homogeneous quadratic (or higher degree) equations in three variables and also verify special cases of a conjecture of Chowla for averages of "aperiodic" multiplicative functions evaluated on homogenous polynomials in two variables.

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