Sharpening Borel's result in Diophantine approximation

In this paper, we refine Borel’s 1903 result in Diophantine approximation by providing sharper bounds for the minimum of three consecutive approximation coefficients ${\mathrm{\Theta }}_n(x)$, defined for any real number $x$ with regular continued fraction (RCF) expansion $x=[0;a_1,a_2,\dots ]$ as ${\mathrm{\Theta }}_n=q_n^2\mid x-\frac{p_n}{q_n}\mid$. 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 $n\in \mathbb{N}$, $$\min \{{\mathrm{\Theta }}_{n-1}(x),{\mathrm{\Theta }}_n(x),{\mathrm{\Theta }}_{n+1}(x)\}\le \frac{1}{\sqrt{5}}\mathrm{.}$$ 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 $1\mathrm{/}\sqrt{5}$ can already be improver to $1\mathrm{/}\sqrt{8}$.