The quotient set of k-generalised Fibonacci numbers is dense in Q_p

The quotient set of $A \subseteq \mathbb{N}$ is defined as $R(A) := \{a / b : a,b \in A,\; b \neq 0\}$. Using algebraic number theory in $\mathbb{Q}(\!\sqrt{5})$, Garcia and Luca proved that the quotient set of Fibonacci numbers is dense in the $p$-adic numbers $\mathbb{Q}_p$, for all prime numbers $p$. For any integer $k \geq 2$, let $(F_n^{(k)})_{n \geq -(k-2)}$ be the sequence of $k$-generalized Fibonacci numbers, defined by the initial values $0, 0, \ldots , 0, 1$ ($k$ terms) and such that each term afterwards is the sum of the $k$ preceding terms. We use $p$-adic analysis to generalize Garcia and Luca's result, by proving that the quotient set of $k$-generalized Fibonacci numbers is dense in $\mathbb{Q}_p$, for any integer $k \geq 2$ and any prime number $p$.