Haudorff dimension and Liouville numbers
In this post we present a proof of the fact the the set of Liouville numbers contained in $[0,1]$ has Hausdorff dimension $0$. Why is this result interesting, apart from finding explicitly the Hausdorff dimension of a particular set? Regarding Lebesgue measure, it is not obvious whether there exists an uncountable set with measure zero. Lioville numbers serve as such example. However, Liouville numbers even serve as an example of an uncountable set with Hausdorff dimension zero, and this is much stronger. That is because every set of dimension zero has measure zero, whereas for example, another common reference of an uncountable set with measure $0$, the Cantor set, has positive Hausdorff dimension ($\ln 2/\ln 3$). Thus the set of Lioville numbers, is in a sense, way smaller than the Cantor set. One could also say that Liouville numbers are example of the smallest possible uncountable sets among the sets with measure zero. You can look here, for another construction of u...