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 $(\mathbb{ T}^\omega, T)$.
Categories: math.NT math.DS