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.