Uniform convergence criterion for non-harmonic sine series

We show that for a nonnegative monotone sequence $\{c_k\}$ the condition $c_kk\to 0$ is sufficient for uniform convergence of the series $\sum_{k=1}^{\infty}c_k\sin k^{\alpha} x$ on any bounded set for $\alpha\in (0,2)$, and for an odd natural $\alpha$ it is sufficient for uniform convergence on the whole $\mathbb{R}$. Moreover, the latter assertion still holds if we replace $k^{\alpha}$ by any polynomial in odd powers with rational coefficients. On the other hand, in the case of an even $\alpha$ it is necessary that $\sum_{k=1}^{\infty}c_k<\infty$ for convergence of the mentioned series at the point $\pi/2$ or at the point $2\pi/3$. Consequently, we obtain uniform convergence criteria. Besides, the results for a natural $\alpha$ remain true for sequences from more general RBVS class.