Freiman homomorphisms on sparse random sets

TL;DR

This paper improves bounds on the size of sparse random sets in inite cyclic groups for which Freiman homomorphisms can be extended, showing the threshold is tight up to constants.

Contribution

The authors establish a sharper bound for the extension of Freiman homomorphisms on sparse random sets, improving previous results and approaching optimality.

Findings

Extended the bound for Freiman homomorphisms to $CN^{-2/3}(\,log N)^{1/3}$

Proved the new bound is essentially optimal up to a constant factor

Demonstrated high-probability extension property for random subsets of inite cyclic groups.

Abstract

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.