On Liouville decompositions in local fields

In 1962 Erd\H{o}s proved that every real number may be decomposed into a sum of Liouville numbers. Here we consider more general functions which decompose elements from an arbitrary local field into Liouville numbers. Several examples and applications are given. As an illustration, we prove that for any real numbers $\alpha _{1},\thinspace \alpha _{2},\ldots ,\thinspace \alpha _{N}$, not equal to 0 or 1, there exist uncountably many Liouville numbers $\sigma$ such that $\alpha _{1}^{\sigma },\thinspace \alpha _{2}^{\sigma },\thinspace \ldots ,\thinspace \alpha _{N}^{\sigma }$ are all Liouville numbers.