I was considering finite lists of integers, and I defined a notion of "density". That is a list of integers $X$ has density $p$ if $G$ is the size of all unique sums of subsets of $X$ and $p = |G|/|X|$.
So as an example if $X = \lbrace 1,1,1,1 \rbrace $
Then the possible unique sums we can have are $0,1,2,3,4$ so the density of $X$ will be $\frac{5}{4}$
Now fixing a size of $X$ to be n, it is easy to create infinite families of primitive $X$ (Meaning none is a non 1/-1 integer multiple of another list other than itself) of that size that have density 2.
Such as:
Consider $E = 1,2,3,4,5....$
Then consider the list of (n-1) E's followed by $2E-1$.
All the lists(one for each $E$) has density 2, and are all primitive.
But how does one derive a family of such primitive lists that have density less than 2.
After toying with some examples it quickly becomes hard, and it's not clear if building such families is even possible.