A noncommutative extension of Mahler’s theorem on interpolation series

In this paper, we prove an extension of Mahler’s theorem on interpolation series, a celebrated result of p -adic analysis. Mahler’s original result states that a function from N to Z is uniformly continuous for the p -adic metric d p if and only if it can be uniformly approximated by polynomial functions. We prove the same result for functions from a free monoid A ∗ to Z , where d p is replaced by the pro- p metric, the profinite metric on A ∗ defined by p -groups.