On the p-adic denseness of the quotient set of a polynomial image

The \emph{quotient set}, or \emph{ratio set}, of a set of integers $A$ is defined as \begin{equation*} R(A) := \left\{a/b : a,b \in A,\; b \neq 0\right\} . \end{equation*} We consider the case in which $A$ is the image of $\mathbb{Z}^+$ under a polynomial $f \in \mathbb{Z}[X]$, and we give some conditions under which $R(A)$ is dense in $\mathbb{Q}_p$. Then, we apply these results to determine when $R(S_m^n)$ is dense in $\mathbb{Q}_p$, where $S_m^n$ is the set of numbers of the form $x_1^n + \cdots + x_m^n$, with $x_1, \dots, x_m \geq 0$ integers. This allows us to answer a question posed in [Garcia \textit{et~al.}, $p$-adic quotient sets, Acta Arith.~\textbf{179}, 163--184]. We end leaving an open question.