Given the Calkin-Wilf enumeration of $\mathbb Q$, $\{q_n\}_{n=1}^\infty$, one can define a cover of the positive rationals: $\left\{ \left(q_n-\frac{1}{2^n}, q_n+\frac{1}{2^n}\right) \right\}_{n=1}^\infty$. This will not be a cover of $(0,\infty)$. This has measure less than $2\epsilon$.
Then is $\pi$ in the cover? What about the golden ratio? The golden ratio with its poor rational approximations seems like it wouldn't be but then again a proof would have to say something about the enumeration. For example, its inverse is not although it has similar properties.
Finally, what is the Lebesgue measure of $\bigcup_n U_n$?
I've seen similar questions but I'm more interested in when an explicit enumeration is given. This enumeration has a nice description in terms of the number of "hyperbinary" representations of $n$, which I hope would help in answering these questions.
