p-Adic Denseness of Members of Partitions of \(\pmb {\mathbb {N}}\) and Their Ratio Sets

Abstract

The ratio set of a set of positive integers A is defined as \(R(A) := \{a / b : a, b \in A\}\). 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 \(\mathbb {Q}_p\) has also been investigated. Let \(A_1, \ldots , A_k\) be a partition of \(\mathbb {N}\) into k sets. We prove that for all prime numbers p but at most \(\lfloor \log _2 k \rfloor \) exceptions at least one of \(R(A_1), \ldots , R(A_k)\) is dense in \(\mathbb {Q}_p\). Moreover, we show that for all prime numbers p but at most \(k - 1\) exceptions at least one of \(A_1, \ldots , A_k\) is dense in \(\mathbb {Z}_p\). Both these results are optimal in the sense that there exist partitions \(A_1, \ldots , A_k\) having exactly \(\lfloor \log _2 k \rfloor \), 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.