M\"obius disjointness conjecture for Furstenberg's flow on $\mathbb{T}^\omega$ in short intervals

Abstract

Furstenberg's flow on the infinite-dimensional torus $\mathbb{T}^\omega$ is defined by \[ T (x_1, x_2, \ldots, x_\nu, \ldots) = (x_1 + \alpha, x_2 + h(x_1), \ldots, x_\nu + h(x_1 + (\nu-2)\beta), \ldots) \] with $\alpha\in \mathbb{R}$ satisfying certain diophantine conditions, $\beta\in \mathbb{R}\backslash\mathbb{Q},$ and $h: \mathbb{R}\to \mathbb{R}$ being $1$-periodic and analytic. This flow is irregular in the sense that its Birkhoff average does not exist for some $x\in \mathbb{T}^\omega$, and it is a generalization of Furstenberg's irregular flow on $\mathbb{T}^2$. The main result of this paper is that the M\"{o}bius Disjointness Conjecture of Sarnak holds for the above flow $(\mathbb{T}^\omega, T)$ in short intervals $(N-M, N]$ with $N^{5/8+\varepsilon} \leqslant M\leqslant N$.