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As @quiz says, for $k$ fixed, 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/

As @quiz says, for $k$ fixed, there exists $k$ integers $a_1,\dots,a_k\in [1,N]$ with $\min|a_i+a_j-(a_r+a_s)|\sim N/k^2$.
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 $\delta\le \frac 1{k(k-2\sqrt k)}$.