In a paper by Simons & Weger in 2005, they talk about approximating the value of $\delta = \dfrac{\log3}{\log2}$ through continued fraction theory using integers $p$ and $q$.
Specifically, they state in (4.2):
"Recall that we denote by $p_n/q_n$ the $n$th convergent to $\delta$. Continued fraction theory shows that convergents are best approximations, i.e. any other approximation with smaller denominator is worse.
When I read through the Wikipedia article on continued fractions, I am finding the following property (see Theorem 5, Corollorary 1) in section 7.2 Some Useful Theorems:
"A convergent is nearer to the limit of the continued fraction than any fraction whose denominator is less than that of the convergent."
Let:
- $p_n, q_n$ be the $n$th convergent for $\delta$
- $K,L$ be positive integers where $2^{K+L} > 3^K$
- $|x|$ be the absolute value of $x$
Does it now follow that:
$$|(K+L) - K\delta| \ge |p_n - q_n\delta|$$
I am asking because I am trying to understand the implications of Theorem 5, Corollary 1 and trying to understand the reasoning behind Lemma 10 in the paper by Simons & Weger.
My assumption would be that:
$$|\frac{K+L}{K} - \delta| \ge |\frac{p_n}{q_n} - \delta|$$
But it is not clear to me that this would imply $|(K+L) - K\delta| \ge |p_n - q_n\delta|$. Since $K \le q_n$, it is not clear to me that dividing the one side by $K$ will necessarily still be greater than dividing the other side by $q_n$.
I also have my doubts about how absolute values affect the equation.
Does Theorem 5, Corollary 1 from Wikipedia, combined with my assumptions, lead to this conclusion:
$$|(K+L) - K\delta| \ge |p_n - q_n\delta|$$
If yes, could you show the details of how it follows (either by formally stating the definition of Theorem 5, Corollary 1 or showing how the result derives from a formal definition).