Sarnak’s Conjecture: What’s New

Abstract

An overview of last seven years results concerning Sarnak's conjecture on Möbius disjointness is presented, focusing on ergodic theory aspects of the conjecture.1 Most often, however not always, T will be a homeomorphism. 2 µ stands for the arithmetic Möbius function, see next sections for explanations of notions that appear in Introduction.3 To be compared with Möbius Randomness Law by Iwaniec and Kowalski [95], page 338, that any "reasonable" sequence of complex numbers is orthogonal to µ.

On disjointness properties of some parabolic flows

“…Since the flows appearing in Theorems 1.1 and 1.2 are totally ergodic (in fact, they are known to be mixing, see respectively [35] for Arnol'd flows and [28] for time changes of horocycle flows), in view of the so called Kátai-Bourgain-Sarnak-Ziegler criterion for orthogonality in [5] (recalled in Section 9), our results in particular imply that Möbius disjointness holds for all smooth time-changes in B + (M ) of horocycle flows and for all uniquely ergodic models of Arnol'd flows (see Section 9 for more details). In particular, Theorem 1.1 answers positively Ratner's question on Möbius disjointness (Question 7 in [11]).…”

“…Corollary 9.2 brings the positive answer to M. Ratner's question (see 7 in [11]). Although, Möbius disjointness itself is known for horocycle flows [5], it remains however an open question whether the assertions of Corollary 9.2 hold for horocycle flows themselves even when u = µ. converge to zero uniformly in x ∈ X (for each h ∈ C(X)) then one can show (considering respectively the sums ( * ) with T −N x and T −1 or with T N x and T ) that we have also convergence for sums of the following type: We do not know if (9.5), (9.6) hold in case of horocycle flows or locally Hamiltonian flows on T 2 .…”

“…We now briefly discuss a relation between Theorems 1.1 and 1.2 with the Möbius disjointness problem [34] which is lately under intensive study, see e.g. survey [11]. If (T t ) is a continuous flow on a compact metric space X then it is called Möbius disjoint if…”

“…Möbius disjointness for horocycle flows has been proved by Bourgain, Sarnak and Ziegler in [5]. A few years ago M. Ratner asked about Möbius disjointness of (smooth) time changes of horocycle flows (see Question 7 in [11]). Since the flows appearing in Theorems 1.1 and 1.2 are totally ergodic (in fact, they are known to be mixing, see respectively [35] for Arnol'd flows and [28] for time changes of horocycle flows), in view of the so called Kátai-Bourgain-Sarnak-Ziegler criterion for orthogonality in [5] (recalled in Section 9), our results in particular imply that Möbius disjointness holds for all smooth time-changes in B + (M ) of horocycle flows and for all uniquely ergodic models of Arnol'd flows (see Section 9 for more details).…”

“…Indeed, Theorem 1.1 not only answers positively Ratner's question on Möbius disjointness, but it also yields uniform convergence (in x ∈ M ) in (1.2). On the other hand, the question of whether uniform convergence holds for horocycle flows themselves remains largely open, see Question 5 in [11]. Thus, this is another instance, where in the non-homogeneous set up one can prove stronger properties than in the homogeneous world.…”