On arithmetic functions orthogonal to deterministic sequences

2021/05/25 by Kanigowski, Adam, Kulaga-Przymus, Joanna, Lemańczyk, Mariusz +1 · 4 citations
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2105.11737

Abstract

We prove Veech's conjecture on the equivalence of Sarnak's conjecture on Möbius orthogonality with a Kolmogorov type property of Furstenberg systems of the M''obius function. This yields a combinatorial condition on the Möbius 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.

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• Furstenberg systems of certain sequences of superpolynomial growth

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