Let $\{\frac{1}{n}\}_{n \in \mathbb N}$ be a sequence in $\mathbb Q$. Now in each interval $(\frac{1}{n+1},\frac{1}{n}]$ define a similar sequence $$ \left\{\frac{1}{n+1}+\frac{\frac{1}{n}-\frac{1}{n+1}}{k_1}\right\}_{k_1 \in \mathbb N} =\left\{\frac{nk_1+1}{nk_1(n+1)}\right\}_{k_1 \in \mathbb N} $$ and in each interval $(\frac{1}{n+1}+\frac{\frac{1}{n}-\frac{1}{n+1}}{k_1+1},\frac{1}{n+1}+\frac{\frac{1}{n}-\frac{1}{n+1}}{k_1}]$ define another sequence, and so on forever
In short devide each interval formed by to two subsequent points of the previous harmonic sequence by another harmonic sequence.
Question: does the set of all points above, i.e. the union of all these sequences (as a set), cover the interval $(0,1] \cap \mathbb Q$ ? i.e. Is there for ervery $q \in (0,1] \cap \mathbb Q$ some $m$-tuple $(n,k_1,k_2,\dots,k_m) \in \mathbb N^{m+1}$ for some $m \in \mathbb N$ such that (I'm not sure of the expression, it can be fatally wrong) $$ q = \frac{1}{n+1} + \frac{\ell(n)}{k_1+1} + \cdots + \frac{\ell(n,k_1,k_2,\dots,k_{m-2})}{k_{m-1}+1} + \frac{\ell(n,k_1,k_2,\dots,k_{m-1})}{k_m} $$ where $\ell (\cdots)$ is the lenght of the interval as function of $(n,k_1,k_2,\dots,k_{m-1})$. I think it is $$ \ell(n,k_1,k_2,\dots,k_{m-1}) = \prod_r^{m-1}\left[\frac{1}{k_r}-\frac{1}{k_r +1}\right] $$
No, the fraction $\frac{2}{5}$ is not in the set. This is because the tuple associated would be $(2,2,2,2,2,...)$ and would not end in a finite number of steps. Indeed, we can see that $\frac{1}{3} < \frac{2}{5} < \frac{1}{2}$, so the $n$ must be $2$. After that we search in the interval $(1/3, 1/2)$ that has length $1/6$.
Let's scale this interval so that it goes to $[0,1]$. Notice that doing this the new sequence ends up being the original harmonic sequence. The transformation that does this is sustracting $1/3$ and multiplying by $6$. The image of $2/5$ under this transformation is:
$$\left(\frac{2}{5} - \frac{1}{3}\right)6 = \frac{6}{15} = \frac{2}{5}.$$
So we would apply the same process, and we would never end. There are more examples like this one.