Irrationality exponent of π and Flint–Hills convergence

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Irrationality exponent of π and Flint–Hills convergence

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Decide whether a series like Σ 1/(nᵃ |sin n|ᵇ) converges by seeing how well fractions approximate π, and see where the bound that settles it comes from.

From spikes to approximations

The Flint–Hills series Σ 1/(n³ sin² n) has sudden spikes. At n = 355 one term is about 25 [0:00]. Each spike is a good rational approximation in disguise: when p/q is a convergent of π, p is very close to qπ, so sin p is tiny [0:23]. The irrationality exponent μ(π) measures how good those approximations can get. Pigeonhole gives μ ≥ 2 [0:51]. The paper's Theorem 1.1 gives the upper bound: for every ν > 2, |π − p/q| ≥ q^(−ν) for all q ≥ Q(ν) [1:16]. This is not the stronger bound c/q² (bounded partial quotients), and Q(ν) is not effective.

The worked case: Flint–Hills

Group n by its nearest multiple qπ. Each group holds at most 4 integers, n ≥ πq/2, and |sin n| ≥ (2/π)‖qπ‖. The series is controlled by Σ 1/(q³‖qπ‖²) [1:46].
Use dyadic blocks K ≤ q < 2K. The points qπ on the circle are d-separated, with d = c(2K)^(1−ν), so their distances are at least d, 2d, 3d… Each block contributes O(K^(2ν−5)) [2:15].
Any ν < 5/2 makes the blocks shrink geometrically, so the series converges.

The general rule (Corollary 5.2): Σ 1/(nᵃ|sin n|ᵇ) converges exactly when a > max{1, b} [3:10]. When a ≤ b, the convergent numerators make the terms at least c·p^(b−a), so for a = b = 2 the terms never tend to 0 and the series diverges [3:37].

Where μ(π) = 2 comes from

The proof works by contradiction [3:59]. Suppose some ν > 2 allows close approximations with unbounded denominators. Pick finitely many of them, with denominators far apart. Interpolation with separated weights (thresholds that do not depend on the centers) gives a nonzero minor Δ_H. Then two estimates collide:

arithmetic floor: after clearing denominators, Δ_H is a nonzero Gaussian integer, so its modulus is ≥ 1 [4:41];
analytic ceiling: repeated Taylor degrees within a group cancel, which forces a saving quadratic in the group size. Rows with large index bring powers of the approximation errors instead [4:51].

The ceiling falls below the floor, so no exponent above 2 survives [5:23]. Move to keep: rewrite sin n as ‖qπ‖, then let an irrationality bound space those distances out, block by block.

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