I had some trouble coming up with a good title for this question. Here is the setup. Suppose you have two infinite sets of (positive real, say) numbers $\{a_k\}$ and $\{b_k\}$ such that the cardinality of the set $A_L$ of $a_k$ smaller than a bound $L \gg 1$ is asymptotic to $L^\alpha,$ and the same is true for the corresponding set $B_L$ set of $b_k$ smaller than $L.$ Now, let's suppose that the $a$s are numbered in increasing order.

The question is: is there some natural/easily checkable condition on the numbering of the $b$s, so that we can say that the cardinality of the set $C_L$ of $c_k = a_k + b_k$ smaller than $L$ has either the same asymptotics as the cardinalities of $A_L, B_L$ (with some different constant), or the same order of growth (for example, this is true if the $b$s are also numbered in increasing order.)

This question really comes from group actions, where the "indices" are group elements, and the $a, b, c$ are the sizes of some features transformed by the group. It looks like something people might have looked at.