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

TL;DR

This paper investigates the threshold probability for a random subset of Z_N to have only trivial Freiman homomorphisms, establishing a phase transition around p = N^{-2/3} to N^{-1/2+\u03b5}.

Contribution

It provides a geometric characterization of linear subsets and determines the probability thresholds for linearity in random subsets of Z_N.

Findings

If p = o(N^{-2/3}), A is not linear with high probability.

If p = N^{-1/2+5}, A is linear with high probability.

Establishes a phase transition in linearity of random subsets.

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.