AI 中文总结
本文解决了弗雷曼-莱夫猜想中$a_{k-2}\geqslant 2k-4$且$a_{k-1}\geqslant 2k-2$的最终难题，完成了该著名受限和集相关猜想的完整证明。
AI 中文摘要
设$A=\{a_{0}, a_{1}, \ldots, a_{k-1}\}$是一个包含$k>7$个整数的集合，满足$0=a_{0}<a_1<\cdots<a_{k-1}$且$\gcd(A)=1$。集合$2^{\wedge}A=\{a+b: a, b\in A, a\neq b\}$称为$A$的受限和集。弗雷曼-莱夫（Freiman-Lev）猜想是与受限和集相关的著名猜想[V.F. Lev，《群中的受限集合加法，I.经典情形》，《伦敦数学会杂志》62(2000)，27-40]。截至目前，当$a_{k-2}\geqslant 2k-4$且$a_{k-1}\geqslant 2k-2$时，弗雷曼-莱夫猜想仍未解决。本文通过解决这一最终且最具挑战性的情形，完成了弗雷曼-莱夫猜想的证明。
英文摘要
Let $A=\{a_{0}, a_{1}, \ldots, a_{k-1}\}$ be a set of $k>7$ integers such that $0=a_{0}<a_1<\cdots<a_{k-1}$ and $\gcd(A)=1$. The set $2^{\wedge}A=\{a+b: a, b\in A, a\neq b\}$ 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 $a_{k-2}\geqslant 2k-4$ and $a_{k-1}\geqslant 2k-2$. In this paper, we complete the proof of the Freiman-Lev conjecture by resolving this final and most challenging case.
