Dense yet scarse: rational numbers.

Question

I have encountered the puzzling fact (just right here on MSE twice: here and here) that the union of intervals of positive length around rational numbers do not cover the real line.

Consider an ordering of the rational numbers $\{r_n\}$ and the intervals belonging to each of them

$$I_n=(r_n-q^{-n},r_n+q^{-n}), \ \ q>3, \ \ n=1,2,\dots$$

The Lebesgue measure of the union of these intervals is surprisingly small:

$$\lambda\left(\bigcup_{n=1}^{\infty}I_n\right)\le\sum_{n=1}^{\infty}\lambda(I_n)=2\sum_{n=1}^{\infty}q^{-n}=\frac{2}{q-1}<1.$$

That is, the union above does not even cover $[0,1]$. So, there are real numbers in the unit interval that are not covered by the union in question. Clearly, the set of the missing reals depend on the ordering we use. OK. But let's take, as an example, the well known classical ordering.

Could anybody point at (construct) an actual real number not in the union belonging to the ordering mentioned?

See How would one go about proving that the rationals are not the countable intersection of open sets? AND Finding an irrational not covered in standard proof that $\mu(\mathbb{Q} \cap [0,1]) = 0$ AND continued — Dave L. Renfro, Mar 14 '17 at 20:35
Constructing a number not in $\bigcup\limits_{k=1}^{\infty} (q_k-\frac{\epsilon}{2^k},q_k+\frac{\epsilon}{2^k})$ AND Showing an element is not in a cover of $\mathbb Q$ (this last question was never answered). — Dave L. Renfro, Mar 14 '17 at 20:37
@DaveL.Renfro: Do you consider my question to be a duplicate of those you mentioned? I yes, then I delete my question. Please let me know. (I have the right to vote to delete posts especially in the case of my own stuff : ) — zoli, Mar 15 '17 at 14:22
My feeling is that it is a duplicate, but rather than deleting your question it would probably be best to edit it so that "Possible Duplicate" is at the top -- see the question "Finding an irrational not covered ..." -- since there is already an answer to your question. (Click the edit answer question to "Finding an irrational ..." to see how to format this.) However, I am not very familiar with policies such as this. — Dave L. Renfro, Mar 15 '17 at 15:46

Dense yet scarse: rational numbers.

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