Distjointness of Mobius from horocycle flows
  We formulate and prove a finite version of Vinogradov's bilinear sum inequality. We use it together with Ratner's joinings theorems to prove that the Mobius function is disjoint from discrete horocycle flows on $\Gamma \backslash SL_2(\mathbb{R}). 
Jean Bourgain, Peter Sarnak and Tamar Ziegler
math.NT math.DS