Möbius disjointness for homogeneous dynamics

Many aspects of the randomness of the prime numbers can be formulated in terms of the Möbius function, and its study has yielded numerous powerful insights into the nature of their distribution. Recently, Sarnak has conjectured a description of its behavior that subsumes many previously known examples: any observable sequence of complex numbers from a zero-entropy topological dynamical system must fail to correlate with the Möbius function. We make use of the full generality of Ratner’s theorems to prove that this conjecture holds for all unipotent translations on homogeneous spaces of real, connected Lie groups, generalizing results of Green-Tao and others.