Diophantine Approximations and the Convergence of Certain Series
Consider two series $$\sum_{n=1}^\infty\frac{\sin^nπθn}{n^α},\quad\sum_{n=1}^\infty\frac{\cos^nπθn}{n^α}.$$ We show that number-theoretical properties of $θ$ have a strong effect on the convergence when $0<α\leq 1$. The complete investigation for $θ\in\mathbb Q$ is given. For irrational $θ$ we prove the result which depends on how well $θ$ can be approximated with rational numbers, i.e. on its irrationality measure. We obtain that if $α>\frac12$ then both series converge absolutely for almost all real $θ$. Finally, we construct such an everywhere dense set of $θ$ that both series diverge when $α\leq 1$.
math.NT↗