In this article we give new characterizations of (N)-denseness of a subset of \(\mathbb {N}\) . Namely, we prove that a set \(D\subset \mathbb {N}\) is (N)-dense if and only if for any infinite subsets A, B of \(\mathbb {N}\) such that \(A\cup B=D\) the quotient set \(R(A;B)=\left\{ \frac{a}{b}:\, a\in A, b\in B\right\} \) is dense in the set of non-negative real numbers. Furthermore, we will discuss multi-dimensional generalizations of two results. The first one, by Bukor, Erdős, Šalát, and Tóth, concerns partitions of (N)-dense sets. The other one, by Bukor and Tóth, gives the relationship between lower and upper asymptotic densities of subsets of \(\mathbb {N}\) and the denseness of their ratio sets in the positive real half-line.
