  I have investigated orders of oscillating sequences motivated by Sarnak's
conjecture in~\cite{JPAMS} and proved that an oscillating sequence of order $d$
is linearly disjoint from affine distal flows on the $d$-torus. One of the
consequences is that an oscillating sequence of order $d$ in the arithmetic
sense is linearly disjoint from affine flows with zero topological entropy on
the $d$-torus. In this paper, I will extend these results to polynomial skew
products on the $d$-torus, that is, given a polynomial skew product on the
$d$-torus, there is a positive integer $m$ such that any oscillating sequence
of order $m$ is linearly disjoint from this polynomial skew product. In
particular, when all polynomials depend only on the first variable, I have that
an oscillating sequence of order $m=d+k-1$ is linearly disjoint from all
polynomial skew products on the $d$-torus with polynomials of degree less than
or equal to $k$. One of the consequences is the linear disjointness for flows
which are automorphisms of the $d$-torus with absolute values of eigenvalues
$1$ plus a polynomial vector and oscillating sequences of order $m$ in the
arithmetic sense. Furthermore, I will prove that an oscillating sequence of
order $d$ is linearly disjoint from minimal mean attractable and minimal
quasi-discrete spectrum of order $d$ flows. Finally, I define and give some
examples of Chowla sequences from our paper~\cite{AJ}.

Dynamical Systems;Number Theory