There can be. Let f(n) be the count of members of the set in question which are at most n. If ε>0 and f(n)=O(n^(1-ε)) then the sum of reciprocals converges. For

There can be. Let f(n) be the count of members of the set in question which are at most n. If ε>0 and f(n)=O(n^(1-ε)) then the sum of reciprocals converges. For denser sets I'd have to think a bit longer about it. It's probably also worth noting that weird density distributions are possible. E.g. one could imagine a kind of oscillation where you include enough values to get the total density up to that point up to some decreasing threshold (e.g. 1-2^-k if we've flip-flopped k times) then omit values to get below a different threshold (e.g. 2^-k if we've flip-flopped k times), and repeat the process indefinitely. The count up to a point n for this construction has the interesting property that it isn't greater in a big-O sense than the count for exponentially sparse sets while also not being less than the count for any sublinear density function in a big-omega sense (using the Knuth interpretation).