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) := { a/b: a, b in A, a not zero }. We consider the case in which A is the image of Z^+ under a polynomial f in Z[X], and we give some conditions under which R(A) is dense in Q_p. Then, we apply these results to determine when R(S^n_m) is dense in Q_p, where S^n_m is the set of numbers of the form x_1^n+...+ x_m^n, with x_1, ...; x_m integers greater or equal than 0. 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.