Bounds on Irrationality Measures and the Flint-Hills Series

Abstract

It is unknown whether the Flint-Hills series $\sum_{n=1}^\infty \frac{1}{n^3\sin^2(n)}$ converges. Alekseyev (2011) connected this question to the irrationality measure of $\pi$, that $\mu(\pi) > \frac{5}{2}$ would imply divergence of the Flint-Hills series. In this paper we established a near-complete converse, that $\mu(\pi) < \frac{5}{2}$ would imply convergence. The associated results on the density of close rational approximations may be of independent interest. The remaining edge case of $\mu(\pi) = \frac{5}{2}$ is briefly addressed, with evidence that it would be hard to resolve.