Detecting approximate sum collisions in two lists.
I believe there is no known algorithm with running time faster than $O(N^2)$, even if you work with integers and ask for a tuple such that $A_a + A_b - (B_c + B_d) = 0$ (i.e., is exactly zero).

There are algorithms with running time $O(N^2)$ and space $O(N)$.  Basically, they iterate through the sum sets $S_A$ and $S_B$ in order, without ever storing the entire sets in memory, by using a priority queue.

There are also algorithms that with running time $O(N^{4/3})$, if the numbers are chosen uniformly at random over some range and if you can choose $N$ appropriately (i.e., if the relationship between $N$ and the size of that range can be chosen appropriately) and if you only care about finding one such 4-tuple (even though many might exist).  I suspect this probably isn't of use to you.