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

The quotient set, or ratio set, of a set of integers $A$ is defined as $R(A) := \left{a/b : a,b \in A,; b \neq 0\right}$\mathrm{.}Weconsiderthecaseinwhich$A$istheimageof$$\mathbb{Z}$^+$underapolynomial$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 et al., $p$-adic quotient sets, Acta Arith. 179, 163-184]. We end leaving an open question.