Partition regularity

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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 an 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 of number theoretic and combinatorial flavor: $(i)$ we give simple necessary and sufficient conditions for the Gowers norms (over $\mathbb{N}$) of a bounded multiplicative function to be zero, $(ii)$ generalizing a classical result of Daboussi and Delange 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 $\lambda\in \mathbb{N}$ such that $16x^2+9y^2=\lambda^2$ and the same holds for the equation $x^2-xy+y^2=\lambda^2$.

Abstract: Furstenberg and Katznelson applied methods of topological dynamics to Ramsey theory, obtaining a density version of the Hales-Jewett partition theorem. Inspired by their methods, but using spaces of ultrafilters instead of their metric spaces, we prove a generalization of a theorem of Carlson about variable words. We extend this result to partitions of finite or infinite sequences of variable words, and we apply these extensions to strengthen a partition theorem of Furstenberg and Katznelson about combinatorial subspaces of the set of words.