On arithmetic functions orthogonal to deterministic sequences - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Advances in Mathematics Année : 2023

On arithmetic functions orthogonal to deterministic sequences

Résumé

We prove Veech's conjecture on the equivalence of Sarnak's conjecture on M\"obius orthogonality with a Kolmogorov type property of Furstenberg systems of the M\"obius function. This yields a combinatorial condition on the M\"obius function itself which is equivalent to Sarnak's conjecture. As a matter of fact, our arguments remain valid in a larger context: we characterize all bounded arithmetic functions orthogonal to all topological systems whose all ergodic measures yield systems from a fixed characteristic class (zero entropy class is an example of such a characteristic class) with the characterization persisting in the logarithmic setup. As a corollary, we obtain that the logarithmic Sarnak's conjecture holds if and only if the logarithmic M\"obius orthogonality is satisfied for all dynamical systems whose ergodic measures yield nilsystems.
Fichier principal
Vignette du fichier
OrthogonalityPP03.pdf (912.79 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03233409 , version 1 (24-05-2021)
hal-03233409 , version 2 (01-06-2021)
hal-03233409 , version 3 (10-09-2021)

Identifiants

Citer

Adam Kanigowski, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Thierry de la Rue. On arithmetic functions orthogonal to deterministic sequences. Advances in Mathematics, 2023, 428, pp.109138. ⟨10.48550/arXiv.2105.11737⟩. ⟨hal-03233409v3⟩
75 Consultations
66 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More