0-1 sequences of the Thue-Morse type and Sarnak's conjecture

TL;DR

This paper proves Sarnak's conjecture for dynamical systems generated by Thue-Morse type sequences and regular Toeplitz sequences, showing spectral measure singularity and orthogonality with the Möbius function, while also providing a counterexample.

Contribution

It establishes Sarnak's conjecture for specific classes of sequences and introduces a non-regular Toeplitz sequence where the conjecture fails.

Findings

Spectral measures are mutually singular for different odd m.

Sarnak's conjecture holds for Thue-Morse type and regular Toeplitz sequences.

Counterexample of a non-regular Toeplitz sequence where the conjecture fails.

Abstract

We show that the images via $z\mapsto z^m$ of the continuous part of the spectral measures of the dynamical systems generated by the 0-1 sequences of the Thue-Morse type are pairwise mutually singular for different odd numbers $m\in\N$. Sarnak's conjecture on orthogonality with the M\"obius function is shown to hold for such dynamical systems. The same conjecture is shown to hold for all systems induced by regular Toeplitz sequences. A non-regular Toeplitz sequence for which Sarnak's conjecture fails is constructed.