Formula omitted]-adic asymptotic properties of constant-recursive sequences

Abstract

In this article we study p-adic properties of sequences of integers (or p-adic integers) that satisfy a linear recurrence with constant coefficients. For such a sequence, we give an explicit approximate twisted interpolation to Zp. We then use this interpolation for two applications. The first is that certain subsequences of constant-recursive sequences converge p-adically. The second is that the density of the residues modulo pα attained by a constant-recursive sequence converges, as α→∞, to the Haar measure of a certain subset of Zp. To illustrate these results, we determine some particular limits for the Fibonacci sequence.