p-adic asymptotic properties of constant-recursive sequences

Rowland, Eric and Yassawi, Reem (2017). p-adic asymptotic properties of constant-recursive sequences. Indagationes Mathematicae, 28(1) pp. 205–220.

DOI: https://doi.org/10.1016/j.indag.2016.11.019

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 ℤ_p. 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 p^α attained by a constant-recursive sequence converges, as α→∞, to the Haar measure of a certain subset of ℤ_p. To illustrate these results, we determine some particular limits for the Fibonacci sequence.

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