Diophantine Approximations and the Convergence of Certain Series

TL;DR

This paper explores how the number-theoretical properties of a parameter influence the convergence of specific sine and cosine series, revealing the role of rationality and irrationality measures.

Contribution

It provides a comprehensive analysis of series convergence based on the rationality and approximation properties of ta, including complete results for rationals and new insights for irrationals.

Findings

Series converge absolutely for almost all ta when lpha > 0.5.

Complete characterization of convergence for rational ta.

Existence of a dense set of ta where series diverge for lpha  1.

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 $\alpha\leq 1$.