Liouville number

A Liouville number is an irrational number $l$ such that for any integer $n>0$ there is a pair of integers $j$ and $k$ such that the inequality

holds. All Liouville numbers are transcendental numbers, but not all transcendental numbers are Liouville numbers. The first example given by Joseph Liouville was a number of the form

$$\sum _{i=1}^{\mathrm{\infty }}\frac{1}{b^{i!}},$$

with an integer $b>1$, specifically $b=10$ (the resulting number is now called Liouville’s constant and is listed in A012245 of Sloane’s OEIS). In base $b$, a number of this form has a representation beginning 0.110001000000000000000001000… where the $i$th instance of the digit 1 is separated from the previous by $i!-(i-1)!-1$ instances of the digit 0.

The set of Liouville numbers is small in measure, having measure 0; in the measure-theoretic setting almost all numbers are not Liouville numbers. However, the set of Liouville numbers is topologically big, as it is residual and in the topological setting almost all numbers are Liouville numbers.