Freiman homomorphisms of random subsets of $\mathbb{Z}_{N}$

Let $A$ be a random subset of $\mathbb{Z}_{N}$ obtained by including each element of $\mathbb{Z}_{N}$ in $A$ independently with probability $p$. We say that $A$ is \emph{linear} if the only Freiman homomorphisms are given by the restrictions of functions of the form $f(x)= ax+b$. For which values of $p$ do we have that $A$ is linear with high probability as $N\to\infty$ ? First, we establish a geometric characterisation of linear subsets. Second, we show that if $p=o(N^{-2/3})$ then $A$ is not linear with high probability whereas if $p=N^{-1/2+\epsilon}$ for any $\epsilon>0$ then $A$ is linear with high probability.