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

We formulate the generalized Sarnak’s M"obius disjointness conjecture for an arbitrary number field $K$, and prove a quantitative disjointness result between polynomial nilsequences $(\mathrm{\Phi }(g(n)\mathrm{\Gamma }){)}_{n\in {\mathbb{Z}}^D}$ and aperiodic multiplicative functions on ${\mathcal{O}}_K$, the ring of integers of $K$. Here $D=[K\text{\hspace{0.17em}⁣}:\mathbb{Q}]$, $X=G\mathrm{/}\mathrm{\Gamma }$ is a nilmanifold, $g\text{\hspace{0.17em}⁣}:{\mathbb{Z}}^D\to G$ is a polynomial sequence, and $\Phi\colon X\to $\mathbb{C}$$ is a Lipschitz function. The proof uses tools from multi-dimensional higher order Fourier analysis, multi-linear analysis, orbit properties on nilmanifold, and an orthogonality criterion of K'atai in ${\mathcal{O}}_K$. 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 ${\mathcal{O}}_K$ to be zero; (3) we provide partition regularity results over $K$ for a large class of homogeneous equations in three variables. For example, for $a,b\in \mathbb{Z}\mathrm{\backslash }\{0\}$, we show that for every partition of ${\mathcal{O}}_K$ into finitely many cells, where $K=\mathbb{Q}(\sqrt{a},\sqrt{b},\sqrt{a+b})$, there exist distinct and non-zero $x,y$ belonging to the same cell and $z\in {\mathcal{O}}_K$ such that $a{x}^2+b{y}^2=z^2$.