The problem can be describe in the following way.

Given a positive integer N, we need to find a methods to generate a set of integers A, so that for any integer 1 <= k <= N, we can find two integers x, y \in A, such that x - y = k.

A example for this is: N=9, we can generate a set A={-3, -2, -1, 0, 3, 6}. This size of A is 6.

We naively prove that the minimal size of |A| = \sqrt(2N). However, we can't find a methods to generate set A that can reach this theoretical minimal. The best methods we find can only reach |A| = 2 \sqrt(N).

Is there any methods that can reach the theoretical minimal \sqrt(2N)? If not, what is the minimal size that any methods can reach?