Almost Countable Spectrum and Logarithmic Sarnak Conjecture

TL;DR

The paper expands the scope of logarithmic Möbius disjointness by introducing almost countable spectrum for zero-entropy topological dynamical systems and proving the logarithmic Sarnak conjecture in this setting. It builds a framework linking ergodic decomposition, Kronecker factors, and maximal pattern entropy, and employs disjointness results with Furstenberg systems of Möbius/Liouville to control correlations. The authors demonstrate the conjecture for broad classes, including group extensions, suspension-flow time-one maps, systems with finite maximal pattern entropy, and bounded tame systems, illustrating the spectral criteria that ensure Möbius disjointness. These results connect spectral properties to arithmetic randomness, providing new avenues for applying dynamical methods to number-theoretic sequences. The work thus broadens the landscape of systems for which logarithmic Sarnak-type disjointness holds and clarifies the structural features (almost countable spectrum, tame behavior) that guarantee it.

Abstract

In this paper, we introduce topological dynamical systems with almost countable spectrum. We prove that the Logarithmic Sarnak Conjecture holds for zero-entropy topological dynamical systems whose spectrum is almost countable. This class includes Anzai skew product on $\mathbb{T}^2$ over a rotation of $\mathbb{T}^1$, time-one maps of continuous suspension flows over rotations, systems with finite maximal pattern entropy, and bounded tame systems.

Theorems & Definitions (55)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Definition 2.1
• Theorem 2.2
• Proposition 2.3