The M öbius Disjointness Conjecture on infinite-dimensional torus

Abstract

Let ${\mathbb{T}}^{\ddot{I}}‰$ be the infinite-dimensional torus, and $T:{\mathbb{T}}^{\ddot{I}}‰\to {\mathbb{T}}^{\ddot{I}}‰$ 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 $\hat{I}\pm \in \mathbb{R},{\hat{I}}^2\in \mathbb{R}\backslash \mathbb{Q},$ and $h:\mathbb{R}\to \mathbb{R}$ being $1$-period and $C^{1+\epsilon }$-smooth. This flow $({\mathbb{T}}^{\ddot{I}}‰,T)$ is distal, and is also irregular in the sense that its Birkhoff average does not exist for all $x\in {\mathbb{T}}^{\ddot{I}}‰$. The main result of this paper is that the M öbius Disjointness Conjecture of Sarnak holds for $({\mathbb{T}}^{\ddot{I}}‰,T)$.