# Expressing real numbers in terms of Liouville numbers

How to express any arbitrary real number $$\alpha$$ as sum of two Liouville numbers?

Recalling a Liouville number is a real number such that $$0<| \alpha - \frac{p}{q}|< \frac{1}{q^n}$$ has infinitely many solutions in $$\frac{p}{q}$$ for all $$n \geq 0.$$

Explicitely $$\sum_{n=1}^{\infty} \frac{a_n}{b^{n!}}$$ such that $$a_n=0,...b-1$$ and $$b\geq 2$$ are Liouville numbers.

Given any arbitrary real number $$\alpha$$ and after considering its decimal expansion how do we construct two Liouville numbers that will add up to $$\alpha$$?

• This is a result of Erdos, here. In that paper he also shows that ever real is the product of two Liouville numbers. – lulu Sep 16 at 13:12
• @lulu Very interesting. Has such a result also been established for the sum ? – Peter Sep 18 at 10:28
• @Peter What do you mean? That paper handles both sum and product. – lulu Sep 18 at 11:21