Equivalent definitions of oscillating sequences of higher orders

An oscillating sequence of order $d$ is defined by the linearly disjointness from all $\{e^{2\pi iP(n)}{\}}_{n=1}^{\mathrm{\infty }}$ for all real polynomials $P$ of degree smaller or equal to $d$. A fully oscillating sequence is defined to be an oscillating sequence of all orders. In this paper, we give several equivalent definitions of such sequences in terms of their disjointness from different dynamical systems on tori.