Higher Order Oscillation and Uniform Distribution

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 $(e^{2\pi i \alpha \beta^{n}g(\beta)}){n\in \N}$,foranon-decreasingtwicedifferentiablefunction$g$ 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 $\big(\alpha \beta^{n}g(\beta)+Q(n)\big){n\in \N}$areuniformlydistributedmodulo$1$.