ON p-ADIC INTERPOLATION IN TWO OF MAHLER’S PROBLEMS

Abstract

AbstractMotivated by the p-adic approach in two of Mahler’s problems, we obtain some results on p-adic analytic interpolation of sequences of integers $(u_n)_{n\geq 0}$ . We show that if $(u_n)_{n\geq 0}$ is a sequence of integers with $u_n = O(n)$ which can be p-adically interpolated by an analytic function $f:\mathbb {Z}_p\rightarrow \mathbb {Q}_p$ , then $f(x)$ is a polynomial function of degree at most one. The case $u_n=O(n^d)$ with $d>1$ is also considered with additional conditions. Moreover, if X and Y are subsets of $\mathbb {Z}$ dense in $\mathbb {Z}_p$ , we prove that there are uncountably many p-adic analytic injective functions $f:\mathbb {Z}_p\to \mathbb {Q}_p$ , with rational coefficients, such that $f(X)=Y$ .