Freiman homomorphisms of random subsets of ${\mathbb{Z}}_N$

Abstract

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.