The M öbius Disjointness Conjecture on infinite-dimensional torus

Let ${\mathbb{T}}^{\omega }$ be the infinite-dimensional torus, and $T:{\mathbb{T}}^{\omega }\to {\mathbb{T}}^{\omega }$ be defined by $$T:(x_1,x_2,\dots ,x_k,\dots )\mapsto (x_1+\alpha ,x_2+h(x_1),\dots ,x_k+h(x_1+(k-2)\beta ),\dots )$$ with $\alpha \in \mathbb{R},\beta \in \mathbb{R}\mathrm{\backslash }\mathbb{Q},$ and $h:\mathbb{R}\to \mathbb{R}$ being $1$-period and $C^{1+\epsilon }$-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 öbius Disjointness Conjecture of Sarnak holds for $({\mathbb{T}}^{\omega },T)$.