$p$-adic asymptotic properties of constant-recursive sequences

TL;DR

This paper investigates the $p$-adic properties of constant-recursive sequences, providing explicit interpolations and analyzing their residue distributions modulo powers of primes, with applications to Fibonacci numbers.

Contribution

It introduces an explicit approximate twisted interpolation for such sequences in $\\mathbb{Z}_p$ and studies their $p$-adic convergence and residue density behavior.

Findings

Certain subsequences converge $p$-adically.

Residue densities approach Haar measure.

Results illustrated with Fibonacci sequence.

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