On spectral disjointness of powers for rank-one transformations and Möbius orthogonality

By el Houcein el Abdalaoui, Mariusz Lemanczyk, Thierry de la Rue. 2013

66 citations recorded by the source corpus.

Abstract We study the spectral disjointness of the powers of a rank-one transformation. For a large class of rank-one constructions, including those for which the cutting and stacking parameters are bounded, and other examples such as rigid generalized Chaconʼs maps and Katokʼs map, we prove that different positive powers of the transformation are pairwise spectrally disjoint on the continuous part of the spectrum. Our proof involves the existence, in the weak closure of { U T k : k ∈ Z } , of “sufficiently many” analytic functions of the operator U T . Then we apply these disjointness results to prove Sarnakʼs conjecture for the (possibly non-uniquely ergodic) symbolic models associated to these rank-one constructions: All sequences realized in these models are orthogonal to the Mobius function.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8389ba2eebc9c6bffec4100d27992443a7b0ce7d

Citations in the corpus

• Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals (2015, unknown): “This assertion is already present in [8], see also [2].”