Popular differences and generalized Sidon sets

For a subset $A\subseteq [N]$, we define the representation function $r_{A-A}(d):=\mathrm{\#}\{(a,a^{\mathrm{\prime }})\in A\times A:d=a-a^{\mathrm{\prime }}\}$ and define $M_D(A) := \max_{1 \leq d < D} r_{A-A}(d)$for$D>1$. We study the smallest possible value of $M_D(A)$ as $A$ ranges over all possible subsets of $[N]$ with a given size. We give explicit asymptotic expressions with constant coefficients determined for a large range of $D$. We shall also see how this problem connects to a well-known problem about generalized Sidon sets.