$p$-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 ${\mathbb{Z}}_p$. 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.

16 pages, 2 figures

References in corpus (3)

• Automatic congruences for diagonals of rational functions
• A Case Study in Meta-AUTOMATION: AUTOMATIC Generation of Congruence AUTOMATA For Combinatorial Sequences
• Profinite automata

Cited by in corpus (2)

• Profinite automata
• Ultimate periodicity problem for linear numeration systems