Let $\delta >0$. Take $\theta \in [0,1]-\mathbb{Q},$ let $\lbrace \frac{m_k}{n_k}\rbrace$ be the sequence of principal convergents to $\theta$, obtained from the continued fraction representation $\theta =[0; a_1, a_2,...],$ where $\sup a_k <\infty$. How fast do the $n_k$ grow?
I have $\log n_k =A_k (1+\delta)^k,$
and I want to know if I can show that $A_k \searrow 0$. Treating $k, n_k$ as real variables, I have $\lim_{k\rightarrow \infty}A_k=\lim_{k \rightarrow \infty}\frac{\log n_k}{(1+\delta )^k}.$ I cannot go further since I don't know how fast the sequence $\lbrace n_k \rbrace$ grows. Using L'Hoptial's Rule I get $\lim_{k \rightarrow \infty}A_k=\lim _{k \rightarrow \infty}\frac{n_k\cdot \frac{\text{d}}{\text{dx}}n_k}{(1+\delta)^k \log(1+\delta)}$
Are there any insights as to whether $\lim A_k$ exists? Any ideas about how fast $n_k$ grows?