$p$-Adic quotient sets: linear recurrence sequences with reducible characteristic polynomials

TL;DR

This paper investigates the density of quotient sets of linear recurrence sequences in p-adic numbers, focusing on sequences with reducible characteristic polynomials, extending previous results on irreducible cases.

Contribution

It classifies primes for which the quotient set of certain linear recurrence sequences with reducible characteristic polynomials is dense in algebraic number fields.

Findings

Identifies conditions under which the quotient set is dense in algebraic fields.

Extends previous work from irreducible to reducible characteristic polynomials.

Provides a classification of primes related to the density of quotient sets.

Abstract

Let $(x_n)_{n\geq0}$ be a linear recurrence sequence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In 2017, Sanna posed an open question to classify primes $p$ for which the quotient set of $(x_n)_{n\geq0}$ is dense in $\mathbb{Q}_p$. In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \alpha$, where $\alpha$ is a Pisot number and if $p$ is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb{Q}_p$, then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb{Q}_p$. In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb{Q}$.