$p$-Adic quotient sets: linear recurrence sequences

Let $(x_n{)}_{n\ge 0}$ be a linear recurrence 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 $a_k\mathrm{\ne }0$. In [`The quotient set of $k$-generalised Fibonacci numbers is dense in ${\mathbb{Q}}_p$’, \emph{Bull. Aust. Math. Soc.} $\mathbf{\text{96}}$ (2017), 24-29], 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 this article, we find a sufficient condition for denseness of the quotient set of the $k$th-order linear recurrence $(x_n{)}_{n\ge 0}$ satisfying $x_n=a_1{x}_{n-1}+a_2{x}_{n-2}+\cdots +a_k{x}_{n-k}$ for all integers $n\ge k$ with initial values $x_0=\cdots =x_{k-2}=0,x_{k-1}=1$, where $a_1,\dots ,a_k\in \mathbb{Z}$ and $a_k=1$. We show that given a prime $p$, there exist infinitely many recurrence sequences of order $k\ge 2$ so that their quotient sets are not dense in ${\mathbb{Q}}_p$. We also study the quotient sets of linear recurrence sequences with coefficients in some arithmetic and geometric progressions.