On the Correlation of the Moebius Function with Rank-one Systems

We explore the ‘Moebius disjointness property’ in the special context of rank-one transformations and verify this phenomenon for many of the ‘classical’ models. (0). Introduction and Preliminaries This note is a follow up on [B-S-Z]. Recall that the general problem considered is the orthogonality of the Moebius sequence {μ(n);n ∈ Z+} with orbits of dynamical systems of zero (topological) entropy. To be more precise, as pointed out in [B-S-Z], we consider a uniquely ergodic topological model for the system. Perhaps the simplest class of systems to study in this context are rank-one transformations and even in this class the problem seems highly nontrivial. It was observed in [B-S-Z] that if an orbit f(n) = φ(Tx), n ∈ Z+ is not orthogonal to the Moebius sequence, then there exist arbitrary large X ∈ Z+ and subsets PX ⊂ [p ∈ P; p ∼ X ] (where P stands for the set of the primes) such that |PX | → ∞ and for p, q ∈ PX