Sarnak's M\"obius disjointness for dynamical systems with singular spectrum and dissection of M\"obius flow

Abstract

It is shown that Sarnak's M\"{o}bius 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 M\"{o}bius 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 We further present an unpublished theorem of Veech which is closely related to our main result.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 M\"{o}bius function, the first projection is in the orthocomplement of its Pinsker algebra then Sarnak M\"{o}bius disjointness conjecture holds. Among other consequences, we obtain a simple proof of Matom{\"a}ki-Radziwi{\l}l-Tao's theorem and Matom{\"a}ki-Radziwi{\l}l's theorem on the correlations of order two of the Liouville function.