TITLE: Selecting $k$ integers from an interval $[0, N]$ to maximize the minimum difference between pairwise sums
QUESTION [5 upvotes]: I have an optimization problem where I need to select $k$ integers over the interval $[0, N]$ s.t. I maximize the minimum difference between any pairwise sum of the $k$ integers (where we also include the sum of one selected integer with itself).  For example, if $k = 3$, $N = 3$, and we select the set of integers $(1, 2, 3)$, we have a set of $\frac{1}{2}k(k+1) = 6$ pairwise sums:
$1 + 1 = 2$
$1 + 2 = 3$
$1 + 3 = 4$
$2 + 2 = 4$
$2 + 3 = 5$
$3 + 3 = 6$
Here, the minimum difference between any two pairwise sums is trivially: $(4 - 4) = 0$.  
As a function of $N$, how does one select the optimal set of $k \leq N$ integers?  What happens in the limit where $N \to \infty$?  I'd also be interested to understand how the number of optimal subsets of $k$ integers scales as $N$ becomes large. 

If we set $N = 100$ and do a computational experiment, we find: 

For $k = 3$: 
Optimal minimum distance between pairwise sums: $33$
Example of subset that achieves the optimal minimum pairwise difference (there are $6$ total): {{0,33,99}} 
All pairwise (and self-) sums for this example subset: {{0,33,66,99,132,198}} 

For $k = 4$: 
Optimal minimum distance between pairwise sums: $16$
Example of subset that achieves the optimal minimum pairwise difference (there are $50$ total): {{0,16,64,96}}
All pairwise (and self-) sums for this example subset: {{0,16,32,64,80,96,112,128,160,192}}

For $k = 5$: 
Optimal minimum distance between pairwise sums: $9$
Example of subset that achieves the optimal minimum pairwise difference (there are $12$ total): {{0,9,36,81,99}}
All pairwise (and self-) sums for this example subset: {{0,9,18,36,45,72,81,90,99,108,117,135,162,180,198}}


Note: From quid's comment, and also given the computational expensive of finding values for $N = 100$ and $k > 5$, it occurs to me that it would be very nice to have some kind of an upperbound, known to the achievable, for the maximum minimum difference for a $k$-sized subset as a function of $k$.  If optimal solutions are dense (I have no reason to suspect that this is the general trend), this could allow for the use of a more efficient probabilistic search procedure to find an optimal "diluted" Sidon subset.  Does anyone have any good ideas for how to construct such an upperbound?

REPLY [3 votes]: As @quiz says,  there exists $k$ integers $a_1,\dots,a_k\in [1,N]$ with $\min|a_i+a_j-(a_r+a_s)|\sim N/k^2$ when $k,\ N/k^2\to \infty$.
Indeed the asymptotic  is sharp. To see this, consider the real numbers $x_j=a_j/N$ and apply the following result of J. Cilleruelo and I. Ruzsa [1]
Theorem: Let $x_1,\dots ,x_k\in [0,1]$ and let $\delta=\min|x_i+x_j-(x_r+x_s)|$. Then
$\delta\le \frac 1{k(k-2\sqrt k)}$.
[1] J. Cilleruelo and I. Ruzsa, "Real and p-adic Sidon sets", Acta Sci. Math. (Szegez)  vol 70, nº 3-4  (2004).  http://www.uam.es/personal_pdi/ciencias/cillerue/