Freiman Homomorphisms of Random Subsets of ZN

Given a set A. Z(N), we say that a function f : A -> Z(N) is a Freiman homomorphism if f(a) + f(b) = f(c) + f(d) whenever a, b, c, d is an element of A satisfy a + b = c + d. This notion was introduced by Freiman in the 1970s, and plays an important role in the field of additive combinatorics. We say that A is linear if the only Freiman homomorphisms are functions of the form f(x) = ax + b. Suppose the elements of A are chosen independently at random, each with probability p. We shall look at the following question: For which values of p = p(N) is A linear with high probability as N ->infinity We show that if p = (2logN -epsilon(N))N-1/3(-2/3), where omega(N)->infinity as N ->infinity, 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.