Sharpening Borel's result in Diophantine approximation

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 $\hat{I}{˜}_n(x)$, defined for any real number $x$ with regular continued fraction (RCF) expansion $x=[0;a_1,a_2,\dots ]$ as $\hat{I}{˜}_n=q_n^2\left|x-\frac{p_n}{q_n}\right|$. 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 $n\in \mathbb{N}$, $$\min \left\{\hat{I}{˜}_{n-1}(x),\hat{I}{˜}_n(x),\hat{I}{˜}_{n+1}(x)\right\}\le \frac{1}{\sqrt{5}}.$$ 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 $1/\sqrt{8}$.