Bounds on Irrationality Measures and the Flint-Hills Series

The Flint-Hills series is the series

and the question of its convergence remains open (Pickover [Pic07]). Alekseyev [Ale11] showed that convergence would imply that $\mu(\pi)\leq\frac{5}{2}$, where $\mu(\pi)$ is the irrationality measure of $\pi$. His argument is easily summarized: if $\mu(\pi)$ is too large, then multiples of $\pi$ are well approximated by integers, $\sin(n)$ becomes exceedingly small, and the individual terms of the sequence grow without bound. The converse is not immediate however, as a small $\mu(\pi)$ leads to a sequence whose terms converge to zero but summed series may still grow without bound.

Our main result is a near-complete converse, that if $\mu(\pi)<\frac{5}{2}$, then the Flint-Hills series converges. This requires inspecting not only the size of the terms, but how frequently good approximations of $\pi$ can occur. The associated results concerning the density of good approximations are elementary to derive and likely known, but the author could not find reference to them in prior literature. They are similar to the bounds in Korhonen [Kor09], where it is shown that the denominators of quadratically good convergents must grow exponentially. Our results refine this in terms of different exponents on the denominator and the irrationality measure. This allows us to establish that the close approximations to $\pi$ are sparse enough that the series must converge.

It is not a complete converse, as the edge case $\mu(\pi)=\frac{5}{2}$ is still consistent with the Flint-Hills series converging or not. We provide evidence that this case cannot be resolved through analysis of the irrationality measure alone: we believe there are two numbers $\alpha_{1}$, $\alpha_{2}$ with both $\mu(\alpha_{i})=\frac{5}{2}$ such that

converges for $i=1$ and diverges for $i=2$. In other words, if we replace the period $\pi$ with a different irrational number, that the series can converge or diverge, and the irrationality measure of the period is not enough information to determine which. We prove in Theorem 3.1 that at least $\alpha_{2}$ exists. But, in the likely event that $\mu(\pi)$ is eventually bounded below $\frac{5}{2}$, this edge case would become irrelevant, and together with our result would imply that the Flint-Hill series converges.

We begin by reviewing the definition of irrationality measure:

Consequently, for any exponent $s>\mu(\alpha)$, there will be only finitely many solutions to $|\alpha-p/q|<1/q^{s}$. To a rational expression $(p,q)$ approximating a real $\alpha$, we associate its approximation exponent $s$,

and to each integer $q$ the minimum approximation exponent across any $(p,q)$.

For a small $\epsilon>0$, there will be infinitely many approximations with $s>\mu(\alpha)-\epsilon$, but we expect these to occur rarely. We call these approximations good:

Sometimes we will simply refer to $q$ as an approximation, in which case we are implicitly taking $p$ to be the integer minimizing $|\alpha-p/q|$. This leads to the related notion of the approximation exponent of each integer $q$:

Our first goal will be to show that good approximations do not cluster close together, in a particular sense:

The first result is a bound on the density of $\epsilon$-approximations, that is, approximations that come close to the irrationality measure $\mu(\alpha)$. They cannot occur with high density, or conversely, the sequence of $q$ with exponent at least $\mu(\alpha)-\epsilon$ must grow quickly.

This bound will allow us to control the contribution of these terms in the Flint-Hill series. There are only a few essential properties of the $|\sin(x)|$ function that we need. We can discuss a generalized Flint-Hill series, built from a function $\mathcal{P}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ with the properties (given an irrational period $\alpha$, and bounds $B_{1}$ and $B_{2}$):

1. 1.

   $\mathcal{P}(x)=\mathcal{P}(x+n\alpha)$, for all $x\in\mathbb{R}^{+},n\in\mathbb{N}$

2. 2.

   If $|x|\leq\alpha/2$, then $B_{1}x\leq\mathcal{P}(x)\leq B_{2}x$.

We call a function with these two properties “sine-like“. Then given exponents $u$ and $v$, the sequence is given by $\mathcal{A}_{u,v}(n)=n^{-u}\mathcal{P}(n)^{-v}$, and its partial sums the series $\mathcal{S}_{u,v}(n)=\sum^{n}_{i=1}\mathcal{A}_{u,v}(i)$. The Flint-Hills series fits this form with $\alpha=\pi$, $B_{1}=\frac{1}{2}$, $B_{2}=1$, $\mathcal{P}=|\sin|$, $u=3$, and $v=2$. [Ale11] gave necessary conditions for the convergence of $\mathcal{S}$:

as for $S$ to converge, $A$ must converge. The only sufficient condition given is the comparatively weak

which in the case of the Flint-Hill series (where $u=3$, $v=2$), gives an (unfulfilled) sufficient condition that $\mu(\pi)<2$. Using Theorem 2.1, we will perform a more careful analysis and give a tight sufficient condition.

In order to show that the series $\sum^{\infty}_{q=1}\mathcal{A}_{u,v}(q)$ converges, we will partition the integers $q$ into different sets based on how accurately they can approximate $1/\alpha$. Before proving the general case, we will demonstrate it with a partitioning of the series into three sets, based on their approximations exponents. This will already give us a nontrivial sufficient condition on the convergence, but not the best condition possible. For the full proof in the next section, we will generalize the technique to finer partitions, and prove the full theorem.

In the Flint-Hills case where $u=3$, $v=2$, this means the sum converges whenever $\mu(\pi)<\frac{3+\sqrt{3}}{2}\approxeq 2.366$. This result required partitioning the integers into a set $S_{1}$ (finitely many $q$’s that are “too good” of an approximation, with $r(q)$ bounded away $\mu(\alpha)$ from above), a set $S_{3}$ ($q$’s that are bad approximations, $r(q)$ at least $\frac{1-u-v}{v}$ less than $\mu(\alpha)$, and can’t diverge no matter how dense they are), and an intermediate set $S_{2}$ that produce large terms in $\mathcal{A}$ but infrequently. We would like to tighten this to match Alekseyev’s bound of $\mu(\alpha)=1+u/v$, and this requires refining the partition $S_{2}$, so that terms with better approximations are appropriately less frequent. In the next section, we prove Theorem 2.1, and apply the above approach to finer partitions.

We now prove that good approximations are not frequently close.

We can now give the proof of our main result.

This result is an almost complete converse of Alekseyev’s Theorem 4, as he has showed that $\mu(\pi)\leq 1+u/v$ is necessary, and we have showed that $\mu(\pi)<1+u/v$ is sufficient. In the event that $\mu(\pi)=1+u/v=5/2$, this will leave unanswered whether or not the Flint Hill series converges. We can show that our bound is in some sense the best possible: there are irrational numbers $\alpha$ with $\mu(\alpha)=1+u/v$ such that the corresponding series can diverge.

The author believes that convergence can also occur:

but proving this would likely involve repeating an analysis similar to the main Theorem 2.5, but more carefully and constructively the whole way through. Theorem 3.1 establishes, at least, that any proof that the original Flint-Hills series converges would need some property of $\pi$ besides its irrationality measure alone.

We established a near converse to the work of Alekseyev. It is widely suspected that $\mu(\pi)=2$, and progress such as that of Zeilberger and Zudilin [ZZ20] has showed that $\mu(\pi)\leq 7.104\dots$, with this bound steadily decreasing over time. If at some point it is established that $\mu(\pi)\leq\frac{5}{2}$, then the convergence Flint-Hills series will be solved. We also established some elementary results on the density of good rational approximations to irrationals in terms of their irrational measures.

This paper would be incomplete without a sincere thank you to Adam Wang, who initially directed the author to this problem, and whose discussions inspired the author to work on this.