  It is known that the M\"obius function in number theory is higher order
oscillating. In this paper we show that there is another kind of higher order
oscillating sequences in the form $(e^{2\pi i \alpha \beta^{n}g(\beta)})_{n\in
\N}$, for a non-decreasing twice differentiable function $g$ with a mild
condition. This follows the result we prove in this paper that for a fixed
non-zero real number $\alpha$ and almost all real numbers $\beta >1$
(alternatively, for a fixed real number $\beta >1$ and almost all real numbers
$\alpha$) and for all real polynomials $Q(x)$, sequences $\big(\alpha
\beta^{n}g(\beta)+Q(n)\big)_{n\in \N}$ are uniformly distributed modulo $1$.
