The M \"obius Disjointness Conjecture on infinite-dimensional torus

TL;DR

This paper proves that Sarnak's M"obius Disjointness Conjecture holds for a specific class of irregular, distal flows on the infinite-dimensional torus, extending the conjecture's applicability.

Contribution

It establishes the M"obius Disjointness Conjecture for a new class of irregular, distal flows on the infinite-dimensional torus.

Findings

M"obius function is disjoint from the flow on the infinite-dimensional torus.

The flow is irregular but still satisfies the conjecture.

Extends the scope of the conjecture to more complex dynamical systems.

Abstract

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