Sarnak's Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow
arXiv:2006.07646
Abstract
It is shown that Sarnak's Möbius orthogonality conjecture is fulfilled for the compact metric dynamical systems for which every invariant measure has singular spectra. This is accomplished by first establishing a special case of Chowla conjecture which gives a correlation between the Möbius function and its square. Then a computation of W. Veech, followed by an argument using the notion of `affinity between measures', (or the so-called `Hellinger method'), completes the proof. We further present an unpublished theorem of Veech which is closely related to our main result. This theorem asserts, if for any probability measure in the closure of the Cesaro averages of the Dirac measure on the shift of the Möbius function, the first projection is in the orthocomplement of its Pinsker algebra then Sarnak Möbius disjointness conjecture holds. Among other consequences, we obtain a simple proof of Matom{ä}ki-Radziwiłl-Tao's theorem and Matom{ä}ki-Radziwiłl's theorem on the correlations of order two of the Liouville function.
23 pages. We present an unpublished theorem of Veech which is closely related to our main result with complete proof. The notion of Quasi-factor plays a key role. In this version, we give more details on the proof of our main result. Comments and suggestions are welcome