Diophantine Approximations and the Convergence of Certain Series

Consider two series $$\sum _{n=1}^{\mathrm{\infty }}\frac{{\sin }^n\pi \theta n}{n^{\alpha }},\sum _{n=1}^{\mathrm{\infty }}\frac{{\cos }^n\pi \theta n}{n^{\alpha }}\mathrm{.}$$ We show that number-theoretical properties of $\theta$ have a strong effect on the convergence when $0<\alpha \le 1$. The complete investigation for $\theta \in \mathbb{Q}$ is given. For irrational $\theta$ we prove the result which depends on how well $\theta$ can be approximated with rational numbers, i.e. on its irrationality measure. We obtain that if $\alpha >\frac{1}{2}$ then both series converge absolutely for almost all real $\theta$. Finally, we construct such an everywhere dense set of $\theta$ that both series diverge when $\alpha \le 1$.