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 uncountable sets with dimension zero. Another interesting aspect of Liouville numbers is that they are $G_\delta$-dense in $[0,1]$ (which is a property stronger than uncountability). Hence Liouville numbers are big in sense of cardinality and topology, yet among the smallest with respect to measure and dimension.

Preliminary notions. We recall the definition of Hausdorff dimension.
For a subset $U$ of a metric space (e.g. real line), define $\operatorname{diam}(U)$ to be the diameter of $U$, i.e. the supremum of distances between any two elements of $U$. If $U$ would be an interval, that is just its length. Now for a set $X$, for $d\ge 0$ (which would take the role of a dimension) and $\delta>0$ define
$$H_\delta^d(X)=\inf\left \{\sum_{i=1}^\infty (\operatorname{diam} U_i)^d: X\subseteq\bigcup_{i=1}^\infty U_i,\ \operatorname{diam} U_i<\delta\right \}.$$
One observes that this is nonincreasing as a function of $\delta$ (smaller $\delta$ - less covers $\{U_n\}_{n\ge 1}$). Thus we define $$\displaystyle\mathcal{H}^d(X)=\sup_{\delta>0}H_\delta^d(X)=\lim_{\delta\to 0}H_\delta^d(X).$$ This turns out to be a measure (defined at least on the Borel $\sigma$-algebra (the one containing all open sets)); it is called $d$-dimensional Hausdorff measure. One may observe that $\mathcal{H}^d(X)$ is nonincreasing as a function of $s$ with values in $[0,\infty]$, and may obtain at most one finite nonzero value (consider for example subset of $[0,1]$).
Define the Hausdorff dimension as the $d$ for which we switch from measure $\infty$ to measure $0$, i.e.
$$\dim_{\operatorname{H}}{(X)}=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}.$$

To the problem. We want to show that the set of Liouville numbers in $[0,1]$ has dimension $0$. Denote this set by $L$. Recall that by definition  $x\in L $ iff for each positive integer $n$ there exist infinitely many positive integers $p,q$, such that $$\left|x-\frac{p}{q}\right|<\frac{1}{q^n}.$$
Pay attention, the usual definition requires existence of at least one such pair $p,q$; it is easy to observe that if there exists at least one pair for each $n$, then there exist infinitely many pairs for each $n$.

So let us unpack what it means for a set to have dimension $0$. Going back to the above definitions (in reverse order) we see that it suffices to show that for any $s>0$ holds $\mathcal{H}^d(X)=0$. This in turn reduces to showing that for $\delta>0$ holds $H_\delta^d(X)=0$. This means that
$$\inf\left \{\sum_{i=1}^\infty (\operatorname{diam} U_i)^d: X\subseteq\bigcup_{i=1}^\infty U_i,\ \operatorname{diam} U_i<\delta\right \}=0$$
Finally, unwrapping the infimum we may summarise the task to be done as follows:

 For any $d>0$, $\delta>0$ and $\varepsilon>0$ there exists a countable collection $\{U_i\}_{i\ge 1}$  such that: $\displaystyle X\subset \bigcup_{i\ge 1}U_i$, $\operatorname{diam}U_i<\delta$ for all $i$ and
$$ \sum_{i=1}^\infty (\operatorname{diam} U_i)^d<\varepsilon.$$ So fix $d>0$, $\delta>0$ and $\varepsilon>0$. We may assume $d<1$. Fix $\displaystyle n>\frac{3}{d},\ q_0>\max\left\{\frac{2}{\delta},\frac{2}{\varepsilon}\right\}$ and consider
$$I_{p,q}:=\left(\frac{p}{q}-\frac{1}{q^n},\frac{p}{q}+\frac{1}{q^n}\right).$$ One observes that (recalling the modified, still equivalent defintion, we stated) $$L\subset \bigcup_{q>q_0}\bigcup_{p=1}^{q-1}I_{p,q}$$ Moreover $\displaystyle\operatorname{diam}I_{p,q}=\frac{2}{q^n}<\frac{2}{q_0}<\delta.$ Finally
 $$\sum_{q>q_0}\sum_{p=1}^{q-1}(\operatorname{diam}I_{p,q})^d\le\sum_{q>q_0}\frac{2^d}{q^{nd}}q\le 2\sum_{q>q_0}\frac{1}{q^{nd-1}}\le 2\sum_{q>q_0}\frac{1}{q^{2}}\le \frac{2}{q_0}<\varepsilon.$$ In the above we used the inequality $$\sum_{k>n}\frac{1}{k^2}\le \frac{1}{n}$$ which could be proven, for example, by estimating the sum from above by $\displaystyle \int_n^\infty\frac{1}{x^2}dx$.

Thus the proof is finished.

You can also check the wikipedia articles for further information on Hausdorff dimension and Liouville numbers.

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