$p$-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 $A_1,\dots ,A_k$ 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 $R(A_1),\dots ,R(A_k)$ 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 $A_1,\dots ,A_k$ is dense in ${\mathbb{Z}}_p$. Both these results are optimal in the sense that there exist partitions $A_1,\dots ,A_k$ 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.