Uniform convergence criterion for non-harmonic sine series

We show that for a nonnegative monotonic sequence $\{c_k\}$ the condition $c_kk\to 0$ is sufficient for the series $\sum_{k=1}^{\infty}c_k\sin k^{\alpha} x$ to converge uniformly on any bounded set for $\alpha\in (0,2)$, and for any odd $\alpha$ it is sufficient for it to converge uniformly on the whole of $\mathbb{R}$. 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 even $\alpha$ it is necessary that $\sum_{k=1}^{\infty}c_k<\infty$ for the above series to converge at the point $\pi/2$ or at $2\pi/3$. As a consequence, we obtain uniform convergence criteria. Furthermore, the results for natural numbers $\alpha$ remain true for sequences in the more general class $\mathrm{RBVS}$.
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