I want to find a set of integers $N$ for which there always exists a pair of numbers $(a, b)$ both $\in N$ such that $a-b = x$ for all $0<x<2^{32}$.
Obviously one possible set N is all the numbers from $0$ to $2^{32}$, but I believe the problem is solvable with a much smaller set, but I am at a loss how to find such a set in a computationally feasible way (ie. not a brute force search).