p-adic denseness of members of partitions of N and their ratio sets

The ratio set of a set of positive integers A is defined as R(A):={a/b:a,b∈A}. The study of the denseness of R(A) in the set of positive real numbers is a classical topic, and, more recently, the denseness in the set of p-adic numbers Qp has also been investigated. Let A1,…,Ak be a partition of N into k sets. We prove that for all prime numbers p but at most ⌊log2k⌋ exceptions at least one of R(A1),…,R(Ak) is dense in Qp. Moreover, we show that for all prime numbers p but at most k−1 exceptions at least one of A1,…,Ak is dense in Zp. Both these results are optimal in the sense that there exist partitions A1,…,Ak having exactly ⌊log2k⌋, respectively, k−1, exceptional prime numbers; and we give explicit constructions for them. Furthermore, as a corollary, we answer negatively a question raised by Garcia et al.