  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.
