Liouville numbers in the non-Archimedean case

Abstract

Basic results about real Liouville numbers are investigated in three non-archimedean settings, referred to as the non-archimedean case, comprising the field of p-adic numbers, the function field completed with respect to the degree valuation and the function field completed with respect to a prime-adic valuation. The result of Erdős that every real number is representable as a sum, and as a product of two real Liouville numbers is shown to hold in the non-archimedean case. The concept of Liouville continued fractions is also considered.