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arXiv · 2607.04421

Sharpening Borel's result in Diophantine approximation

Savita S. Adhin·Ayreena Bakhtawar·Cor Kraaikamp
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Abstract

In this paper, we refine Borel's 1903 result in Diophantine approximation by providing sharper bounds for the minimum of three consecutive approximation coefficients $\Theta_n(x)$, defined for any real number $x$ with regular continued fraction (RCF) expansion $x=[0;a_1,a_2,\dots]$ as $\Theta_n = q_n^2\left| x-\frac{p_n}{q_n}\right|$. Here $\frac{p_n}{q_n}$ is the $n$th RCF convergent of $x$. Borel's result states that for all (irrational) $x$ and all $n\in\mathbb{N}$, $$ \min \left\{ \Theta_{n-1}(x),\Theta_n(x),\Theta_{n+1}(x)\right\} \leq \frac{1}{\sqrt{5}}. $$ We focus on the situation where $a_{n+1}=1$, since otherwise a result by F.~Bagemihl and J.R.~McLaughlin from 1966 implies that the Borel-bound $1/\sqrt{5}$ can already be improver to $1/\sqrt{8}$.

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Savita S. Adhin, Ayreena Bakhtawar, Cor Kraaikamp. 2026-07-05. Sharpening Borel's result in Diophantine approximation. https://arxiv.org/abs/2607.04421

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