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 is an element of A, b not equal 0}. We consider the case in which A is the image of Z(+) under a polynomial f is an element of 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-m(n)) is dense in Q(p), where S-m(n) is the set of numbers of the form x(1)(n) + . . . + x(m)(n), with x(1), . . . , x(m )>= 0 integers. This allows us to answer a question posed in Garcia et al. (2017) [5]. We end leaving an open question. (C) 2018 Elsevier Inc. All rights reserved.