Which irrationals are contained in the Cantor set?
Solution 1:
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!