Not knowing elementary number theory well, I ask this one, which is not very clear to answer, rather I am looking for some results around this question or known theorems. The problem is the following:

The set of prime numbers $\mathbb{P}=2,3,5,7,11...$ generates $\mathbb{N}$ by multiplication. Now I am interested in subsets $S$ of $\mathbb{P}$, which generate a positive fraction of all numbers, that means:

There is $\epsilon>0$ which holds the following equation for all $N \in\mathbb{N}$ big enough: $$\frac{\|span(S)_{\leq N}\|}{N}\geq \epsilon N$$ where $span(S)_{\leq N}$ is the subset of $\mathbb{N}_{\leq N}$, which consists of the numbers, whose prime factors are all elements of $S$.

Clearly $S$ has to be infinite to have this property, but what can one say about $S$ more specificely? Is there any known criterions or examples for $S$ being too small to generate a positive ratio of the natural numbers?