On arithmetic functions orthogonal to deterministic sequences

TL;DR

This paper proves a conjecture linking M"obius orthogonality with a Kolmogorov property of Furstenberg systems, providing a combinatorial criterion for Sarnak's conjecture and extending the characterization to broader classes of functions and systems.

Contribution

It establishes the equivalence of Sarnak's conjecture with a Kolmogorov-type property and characterizes all bounded arithmetic functions orthogonal to certain dynamical systems.

Findings

Proves Veech's conjecture on M"obius orthogonality and Furstenberg systems.

Provides a combinatorial condition equivalent to Sarnak's conjecture.

Shows the equivalence of logarithmic Sarnak's conjecture with M"obius orthogonality for nilsystems.

Abstract

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.