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关于无理数度量与多项式连分数之间的联系

On the Connection Between Irrationality Measures and Polynomial Continued Fractions

Technion - Israel Institute of Technology（以色列理工学院）；Google Inc.（谷歌公司）

Nadav Ben David, Guy Nimri, Uri Mendlovic, Yahel Manor, Carlos De la Cruz Mengual, Ido Kaminer

备注：
57 pages containing 5 appendices and 5 tables
发表信息：
Arnold Math J. 10, 529-566 (2024)
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AI总结：

本文探讨了多项式连分数与无理数度量之间的联系，扩展了阿佩里工作，提出关于PCFs生成的DAs的一般结论和猜想，并应用于基本常数的逼近。

摘要：

具有整数系数的线性递推关系，如生成斐波那契数列的递推关系，已经经过数千年研究，但仍隐藏着新的数学内容。此类递推关系被阿佩里（Apéry）用于证明ζ(3)的无理性，后来被称为阿佩里常数。阿佩里的证明使用了一个特定的线性递推关系，其中包含整数多项式构成的连分数；称为多项式连分数（PCFs）。类似的多项式递推关系被用来证明其他数学常数如π和e的无理性。更一般地，由PCFs生成的序列形成迪奥菲安逼近（DAs），在数学的许多领域如数论中普遍存在。目前还不清楚哪些多项式递推关系能产生有用的DAs，以及它们是否能证明无理性。在这里，我们提出了关于由PCFs生成的DAs的一般结论和猜想。具体而言，我们扩展了阿佩里的工作，超越了他特定选择的PCF，找到在什么条件下PCF能证明无理性或提供高效的DAs。为了提供具体的例子，我们将我们的发现应用于由拉马努金机器算法找到的PCFs，以表示基本常数如π、e、ζ(3)和Catalan常数G。对于每个这样的PCF，我们展示了其收敛率和效率的提取，以及它为基本常数的无理数度量提供的界限。我们进一步基于PCFs提出了新的DAs猜想。我们的发现激励了未来对任何具有整数系数的线性递推关系生成的序列的研究，以帮助开发系统算法来寻找基本常数的DAs。因此，我们的研究可能有助于回答开放问题，如Catalan常数或Riemannζ函数值（例如ζ(5)）的无理性证明。

英文摘要：

Linear recursions with integer coefficients, such as the one generating the Fibonacci sequence, have been intensely studied over millennia and yet still hide new mathematics. Such a recursion was used by Apéry in his proof of the irrationality of $ζ(3)$, later named the Apéry constant. Apéry's proof used a specific linear recursion containing integer polynomials forming a continued fraction; called polynomial continued fractions (PCFs). Similar polynomial recursions prove the irrationality of other mathematical constants such as $π$ and $e$. More generally, the sequences generated by PCFs form Diophantine approximations (DAs), which are ubiquitous in areas of math such as number theory. It is not known which polynomial recursions create useful DAs and whether they prove irrationality. Here, we present general conclusions and conjectures about DAs created from PCFs. Specifically, we generalize Apéry's work, going beyond his particular choice of PCF, finding the conditions under which a PCF proves irrationality or provides an efficient DA. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as $π$, $e$, $ζ(3)$, and the Catalan constant G. For each such PCF, we demonstrate the extraction of its convergence rate and efficiency, as well as the bound it provides for the irrationality measure of the fundamental constant. We further propose new DA conjectures based on PCFs. Our findings motivate future research on sequences created by any linear recursions with integer coefficients, to aid the development of systematic algorithms for finding DAs of fundamental constants. Consequently, our study may contribute to ongoing efforts to answer open questions, such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., $ζ(5)$).

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