Möbius 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 \[ T (x_1, x_2, \ldots, x_ν, \ldots) = (x_1 + α, x_2 + h(x_1), \ldots, x_ν+ h(x_1 + (ν-2)β), \ldots) \] 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 Möbius Disjointness Conjecture of Sarnak holds for the above flow $({\mathbb{T}}^{\omega },T)$ in short intervals $(N-M,N]$ with $N^{5/8+\epsilon }⩽M⩽N$.