Tuesday 3 March 2026, 16:00 - 17:00 in HG 01.028
Cornelis Kraaikamp (TU Delft)
Generalizations of Theorems by Vahlen and Borel for Regular (and Optimal) Continued Fraction Algorithms
Let \(x\in [0,1)\) have (regular) continued fraction expansion \( x = [0;a_1,a_2,\dots] \), and let the \(n\) th regular convergent be obtained by finite truncation, \[ \frac{p_n}{q_n} = [0;a_1,a_2,\dots, a_n], \] where it is assumed that \( p_n, q_n\in \mathbb{Z}, q_n>0 \), and where \(p_n\) and \(q_n\) are relative prime. Let the \(n\)th approximation coefficient \(\Theta_n(x)\) be defined as \[ \Theta_n(x) = q_n^2 \left| x - \frac{p_n}{q_n} \right|. \] In 1895, Vahlen proved his famous theorem in Diophantine approximation, that for \( x \) irrational, and all \( n\in \mathbb{N} \): \[ \min \{ \Theta_{n-1}(x), \Theta_n(x)\} < \tfrac{1}{2}. \] A few years later, in 1903, Borel showed that for \(x\) irrational, and all \( n\in \mathbb{N} \): \[ \min \{ \Theta_{n-1}(x), \Theta_n(x)\} < \tfrac{1}{\sqrt{5}}. \] Recently, in 2015, 2021 and 2021, Vahlen's result was generalized by Jarovslav Hancl and co-authors. Last year, Ayreena Bakhtawar and CK also generalized Vahlen, but in a completely different way. In the last century, Borel's theorem has attracted a lot of attention, and various authors sharpened Borel's result. Hancl and Kit Nair sharpened it in 2024, and Bakhtawar and CK are about to put a paper on ArXiV in which a totally different generalization is given. In this talk we will see that the results of Vahlen and Borel are easy consequences of the natural extension of the (regular) continued fraction expansion, a dynamical system introduced by Hitoshi Nakada (Keio University), which played a major role in the use of continued fraction expansions to find high quality rational approximations to an irrational \(x\). This approach is then refined, leading to generalizations of both the result of Vahlen, and that of Borel. This approach can also be used to refine a Vahlen-type result for Wieb Bosma's Optimal Continued fraction expansion. This is joint work with Ayreena Bakhtawar (IM PAN, Warsaw).