It seems natural to consider $\lim_{q \rightarrow 1^-} \sum_{n \in S} q^n - \sum_{n \in T} q^n$, when it exists, as a way of comparing the sizes of two sets $S,T \subseteq {\bf N}$ that have the same density; for instance, $\{0,2,4,...\}$ and $\{1,3,5,...\}$ both have density 1/2, but the first set might be said to contain "half an element more" than the second, based on the fact that $1-q+q^2-q^3+...$ converges to 1/2 as $q$ goes to 1. Has this (non-Archimedean, non-translation-invariant) refinement of the concept of density, or anything similar to it, been developed anywhere?

Put $|S|_q = \sum_{n \in S} q^n$. Just as $|\{0,2,4,6,...\}|_q-|\{1,3,5,7,...\}|_q \rightarrow 1/2$ as $q \rightarrow 1^-$, it seems empirically that $$|\{0,1,4,9,16,...\}|_q-|\{0,2,6,12,20,...\}|_q \rightarrow 1/2,$$ $$|\{0,1,5,12,22,...\}|_q-|\{0,2,7,15,26,...\}|_q \rightarrow 1/3,$$ and $$|\{0,1,3,6,10,..,\}|_q - \sqrt{2} |\{0,1,4,9,16,...\}|_q \rightarrow \sqrt{2}/2.$$ Is there a place in the literature where results like this can be found, as part of a general setup for measuring sets that "sees" things at a finer level than mere density? To give one last example, racing "evil" versus "odious" numbers, it is easy to prove that $$|\{0,3,5,6,9,10,12,15,...\}|_q-|\{1,2,4,7,8,11,13,14,...\}|_q \rightarrow 0.$$

[ADDED ON MAY 17, 2017: There is now (at least) one article that uses analytic tools to generalize the notion of cardinality of sets of natural numbers: my preprint "One-Dimensional Packing: Maximality Implies Rationality", available at https://arxiv.org/abs/1704.08785 ]