The M öbius Disjointness Conjecture on infinite-dimensional torus

Abstract

Let $\mathbb{T}^ω$ be the infinite-dimensional torus, and $T: \mathbb{T}^ω\to \mathbb{T}^ω$ be defined by \[ T: (x_1, x_2, \dots, x_k, \ldots) \mapsto (x_1 + α, x_2 + h(x_1), \dots, x_k + h(x_1 + (k-2)β), \dots) \] with $α\in \mathbb{R}, β\in \mathbb{R}\backslash\mathbb{Q},$ and $h: \mathbb{R}\to \mathbb{R}$ being $1$-period and $C^{1+\varepsilon}$-smooth. This flow $(\mathbb{T}^ω, T)$ is distal, and is also irregular in the sense that its Birkhoff average does not exist for all $x\in \mathbb{T}^ω$. The main result of this paper is that the M öbius Disjointness Conjecture of Sarnak holds for $(\mathbb{T}^ω, T)$.

Theorems & Definitions (13)

• Theorem 1.1: Möbius Disjointness
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 3.1: Polynomial rate rigidity
• Theorem 3.2: Sub-polynomial measure complexity
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3