lemmaroutine

Flint–Hills sum converges for finite irrationality exponent

∥qα∥≥cq1−ν\|q\alpha\|\ge cq^{1-\nu} with ν<5/2\nu<5/2 makes ∑1/(q3∥qα∥2)\sum 1/(q^3\|q\alpha\|^2) converge
turns an irrationality measure into series convergence
Lemma·series:dyadic·§5 Flint--Hills series and its generalization

Statement (paper)

Let α>0\alpha>0 be irrational. Suppose that, for some c>0c>0 and 1<ν<5/21<\nu<5/2, ∥qα∥≥cq1−ν(q∈Z, q≥1). \|q\alpha\|\ge c q^{1-\nu}\qquad(q\in\ZZ,\ q\ge1). Then ∑q=1∞1q3∥qα∥2<∞. \sum_{q=1}^{\infty}\frac{1}{q^3\|q\alpha\|^2}<\infty.

In plain English

If α>0\alpha>0 is irrational and ∥qα∥≥cq1−ν\|q\alpha\|\ge c q^{1-\nu} for all q≥1q\ge1 with some c>0c>0 and 1<ν<5/21<\nu<5/2, then ∑q≥11/(q3∥qα∥2)\sum_{q\ge1} 1/(q^3\|q\alpha\|^2) converges.

Why it's here

This turns an irrationality-measure bound on α\alpha (here with exponent ν<5/2\nu<5/2) into convergence of the Flint–Hills-type sum, which is the analytic input needed for the Flint–Hills series and its generalization. The exponent ν<5/2\nu<5/2 is what makes the sum converge, and the dyadic grouping of qq is what controls how often qαq\alpha can be close to an integer.
Classified routine:
This follows by a standard spacing argument: the Diophantine bound separates the points qα mod 1q\alpha\bmod 1 in each dyadic block, bounding the block’s contribution by O(N2ν−5)O(N^{2\nu-5}). Summing over dyadic blocks converges when ν<5/2\nu<5/2.

Uses

Nothing else in the paper

Used by

corFlint–Hills series convergescorGeneralized Flint–Hills series convergence criterion

Sources

PaperOpen
Section
§5 Flint--Hills series and its generalization
Page
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Label
series:dyadic
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sections/series.tex:11
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