It is well known that the Cantor set is uncountable. Hence it contains irrationals. What are the 'nice' irrationals in the Cantor set.
Here, I am expecting irrational numbers in the form of square roots of $\frac{1}{n}$, cube roots of $\frac{1}{n}$, or their combinations, or $\pi/n$, $e/n$ ($n\in\mathbb{N}$), or rational powers of $e$, $\pi$, or any such nice form. (In fact we can take a number with ternary expansion with $0$'s and $2$'s, which is not repeating; but I would like to see numbers not in ternary form.)
Define the Jacobi theta function (or whatever this variant is called): $$\theta(q)=\sum_{n=-\infty}^\infty q^{n^2}=1+2\sum_{n=1}^\infty q^{n^2}.$$ Then $$\theta\left(\frac13\right)-1=2\sum_{n=1}^\infty 3^{-n^2}=0.2002000020000002\ldots_3$$ is an irrational number in the middle-thirds Cantor set. You can decide how 'nice' $\theta(1/3)$ is!