On variations of the Liouville constant which are also Liouville numbers

Abstract

Let $\ell$ be the Liouville’s constant, defined as a decimal with a 1 in each decimal place corresponding to $n!$ and 0 otherwise. This number is a classical example of a Liouville number. In this note, we give an optimal condition on the number of replacements of 0’s by 1’s between two consecutive 1’s in the decimal expansion of $\ell$ in order to ensure that this new number is still a Liouville number.