Diophantine Approximations and the Convergence of Certain Series

Alexander Begunts; Dmitry Goryashin

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arXiv · 1506.05746

Diophantine Approximations and the Convergence of Certain Series

Alexander Begunts·Dmitry Goryashin

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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 $\alpha\leq 1$.

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Alexander Begunts, Dmitry Goryashin. 2015-06-18. Diophantine Approximations and the Convergence of Certain Series. https://arxiv.org/abs/1506.05746

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