$p$-adic denseness of members of partitions of $\mathbb{N}$ 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.