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

Abstract

Let $(x_n{)}_{n\ge 0}$ be a linear recurrence sequence of order $k\ge 2$ satisfying $$x_n=a_1{x}_{n-1}+a_2{x}_{n-2}+\cdots +a_k{x}_{n-k}$$ for all integers $n\ge k$, where $a_1,\dots ,a_k,x_0,\dots ,x_{k-1}\in \mathbb{Z},$ with $a_k\ne 0$. In 2017, Sanna posed an open question to classify primes $p$ for which the quotient set of $(x_n{)}_{n\ge 0}$ is dense in ${\mathbb{Q}}_p$. In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \hat{I}\pm$, where $\hat{I}\pm$ 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 $(x_n{)}_{n\ge 0}$ is dense in ${\mathbb{Q}}_p$. In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb{Q}$.

To appear at Canadian Mathematical Bulletin. arXiv admin note: text overlap with arXiv:2207.07084