Recently, I've been studying computable analysis. One of the basic notions is a computable real number, which I will define as any $r \in \mathbb{R}$ which has a computable Cauchy name - a computable, rational sequence $(q_n)_{n \in \mathbb{N}}$ satisfying $\lvert q_n - r \rvert \leq 2^{-n}$ for each $n$.
There is a well-developed theory of computational complexity for computable functions $f: \mathbb{N} \to \mathbb{N}$. It seems natural to try extend this to computable real numbers. For example, say a computable real $r \in \mathbb{R}$ has time complexity $O(g)$, for some function $g: \mathbb{N} \to \mathbb{N}$, if there exists a Cauchy name $(q_n)$ of $r$ which is computable in time $O(g)$ (here, consider $(q_n)$ as a function $\mathbb{N} \to \mathbb{N}$ under some coding of the rationals).
It seems clear then that any rational number $q$ is $O(1)$-computable - just choose the sequence which is constantly equal to $q$. I believe there should be irrationals which are constant-time-computable, but can't think of a quick example. And I'm not sure how one could prove that a real number is $\Omega(g)$ for some $g$. This has clear connections to rate of convergence of a sequence in numerical analysis, but we also have to consider how long it takes to compute terms of the sequence.
Can anyone point me to any work which has been done on such an idea? It seems like something which someone else must have considered. Note I'm not interested in complexity on real-valued functions, but on real numbers themselves.
Edit: to fix the ideas, let us say a real number $r$ is $O(h)$-computable if there exist $O(h)$-computable functions $f, g: \mathbb{N} \to \mathbb{N}$ such that for all $n$, $\left\lvert \frac{f(n)}{g(n)} - r\, \right\rvert \leq 2^{-n}$.
