# Freiman homomorphisms on sparse random sets

Conlon, D ; Gowers, William Timothy (2017-02-03)

Article

A result of Fiz Pontiveros shows that if \$A\$ is a random subset of \$\mathbb{Z}_N\$ where each element is chosen independently with probability \$N^{-1/2+o(1)}\$, then with high probability every Freiman homomorphism defined on \$A\$ can be extended to a Freiman homomorphism on the whole of \$\mathbb{Z}_N\$. In this paper we improve the bound to \$CN^{-2/3}(\log N)^{1/3}\$, which is best possible up to the constant factor.