Sarnak's Conjecture for nilsequences on arbitrary number fields and applications

Abstract

We formulate the generalized Sarnak's Möbius disjointness conjecture for an arbitrary number field K, and prove a quantitative disjointness result between polynomial nilsequences (Φ(g(n)Γ))n∈ZD and aperiodic multiplicative functions on OK, the ring of integers of K. Here D=[K:Q], X=G/Γ is a nilmanifold, g:ZD→G is a polynomial sequence, and Φ:X→C is a Lipschitz function. This result, being a generalization of a previous theorem of the author in [44], requires a significantly different approach, which involves with multi-dimensional higher order Fourier analysis, multi-linear analysis, orbit properties on nilmanifold, and an orthogonality criterion of Kátai in OK.We also use variations of this result to derive applications in number theory and combinatorics: (1) we prove a structure theorem for multiplicative functions on K, saying that every bounded multiplicative function can be decomposed into the sum of an almost periodic function (the structural part) and a function with small Gowers uniformity norm of any degree (the uniform part); (2) we give a necessary and sufficient condition for the Gowers norms of a bounded multiplicative function in OK to be zero; (3) we provide partition regularity results over K for a large class of homogeneous equations in three variables. For example, for a,b∈Z﹨{0}, we show that for every partition of OK into finitely many cells, where K=Q(a,b,a+b), there exist distinct and non-zero x,y belonging to the same cell and z∈OK such that ax2+by2=z2.