Bounds on Irrationality Measures and the Flint-Hills Series

TL;DR

This paper explores the relationship between the irrationality measure of pi and the convergence of the Flint-Hills series, establishing near-complete conditions linking the two and discussing the challenging edge case.

Contribution

It provides a near-complete converse to Alekseyev's result, connecting the irrationality measure of pi to the series' convergence, and analyzes the density of rational approximations.

Findings

If μ(π) < 5/2, the series converges.

If μ(π) > 5/2, the series diverges.

The edge case μ(π) = 5/2 remains unresolved and difficult to determine.

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.