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 ∈ A, b = 0} . We consider the case in which A is the image of Z+ under a polynomial f ∈ Z[X], and we give some conditions under which R(A) is dense in Qp. Then, we apply these results to determine when R(Snm) is dense in Qp, where Snm is the set of numbers of the form xn1 + ··· + xn m, with x1, ..., xm ≥ 0 integers. This allows us to answer a question posed in Garcia et al. (2017) [5]. We end leaving an open question