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 .

Abstract. We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for "typical" choices of Hardy field functions a(t) with polynomial growth, the averagesconverge in the mean and we determine their limit. For example, this is the case if a(t) = t 3/2 , t log t, or t 2 + (log t) 2 . Furthermore, if {a1(t), . . . , a ℓ (t)} is a "typical" family of logarithmico-exponential functions of polynomial growth, then for every ergodic system, the averages

Abstract. We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form {l 1 p, l 2 p, . . . , l k p}. We then derive several multiple recurrence results and combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemerédi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that for all ε > 0 and every subset of the integers Λ the set

Abstract. We prove mean convergence, as N → ∞, for the multiple ergodic averages

The Möbius disjointness conjecture of Sarnak states that the Möbius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness argument and the key ingredient is a structural result for measure preserving systems naturally associated with the Möbius and the Liouville function. We prove that such systems have no irrational spectrum and their building blocks are infinite-step nilsystems and Bernoulli systems. To establish this structural result we make a connection with a problem of purely ergodic nature via some identities recently obtained by Tao. In addition to an ergodic structural result of Host and Kra, our analysis is guided by the notion of strong stationarity which was introduced by Furstenberg and Katznelson in the early 90's and naturally plays a central role in the structural analysis of measure preserving systems associated with multiplicative functions.