A problem I'm investigating concerns a combinatorial class in which the 'atoms' have irrational sizes. It seems likely that this is something that has been considered before, but I haven't been able to find it in the literature.

As a simple example, consider the bivariate generating function $f(x,y)=(1-x-y)^{-1}$ for sequences of two different types of objects ($X$ and $Y$). Now suppose that the size of an $X$ object is $a$ and the size of a $Y$ object is $b$. If $a$ and $b$ are integers, then $f(z^a,z^b)$ enumerates sequences of a given size, and if $a/b$ is rational, we can get the enumeration by multiplying by the least common denominator.

What is the theory when $a/b$ is irrational? Consider $g(z)=f(z,z^\sqrt{2})$. This function is no longer a formal power series in $z$. However, singularity analysis of $g(z)$ seems to deliver the 'right' answer for the 'growth rate', as might be expected by continuity. On the other hand, the concept of growth rate in this case has to be modified a bit, to be something like $\lim_{t \to \infty} {S_{[t,t+1]}}^{1/t}$, where $S_I$ is the number of sequences whose sizes lie in the interval $I$.

Where can I find this kind of thing discussed?