Liouville Number - Structure of The Set of Liouville Numbers

Structure of The Set of Liouville Numbers

For each positive integer n, set

\begin{align} U_n & =\bigcup\limits_{q=2}^\infty\bigcup\limits_{p=-\infty}^\infty \left\{ x \in \mathbb R : 0< \vert x- \frac{p}{q} \vert < \frac{1}{q^{n}}\right\} \\ & = \bigcup\limits_{q=2}^\infty\bigcup\limits_{p=-\infty}^\infty \left(\frac{p}{q}-\frac{1}{q^n},\frac{p}{q}+\frac{1}{q^n}\right) \setminus \left\{\frac{p}{q}\right\}\end{align}.

The set of all Liouville numbers can thus be written as .

Each is an open set; as its closure contains all rationals (the {p/q}'s from each punctured interval), it is also a dense subset of real line. Since it is the intersection of countably many such open dense sets, is comeagre, that is to say, it is a dense Gδ set.

Along with the above remarks about measure, it shows that the set of Liouville numbers and its complement decompose the reals into two sets, one of which is meagre, and the other of Lebesgue measure zero.

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