A proof of the Freiman-Lev conjecture

Abstract

Let $A=\{a_0,a_1,\dots ,a_{k-1}\}$ be a set of $k>7$ integers such that $0=a_0<a_1<\cdots <a_{k-1}$ and $\gcd (A)=1$. The set $2^{\wedge }A=\{a+b:a,b\in A,a\ne b\}$ is called the restricted sumsets of $A$. Freiman-Lev conjecture is a well-known conjecture which related to restricted sumsets [V.F. Lev, Restricted set addition in groups, I. The classical setting, J. London Math. Soc. 62(2000), 27-40]. Up to now, Freiman-Lev conjecture is still open for all $a_{k-2}⩾2k-4$ and $a_{k-1}⩾2k-2$. In this paper, we complete the proof of the Freiman-Lev conjecture by resolving this final and most challenging case.