Diophantine Approximations and the Convergence of Certain Series

Abstract

Consider two series $$\sum_{n=1}^\infty\frac{\sin^n\pi\theta n}{n^\alpha},\quad\sum_{n=1}^\infty\frac{\cos^n\pi\theta n}{n^\alpha}.$$ We show that number-theoretical properties of $\theta$ have a strong effect on the convergence when $0<\alpha\leq 1$. 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 $\alpha>\frac12$ 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 $\alpha\leq 1$.