刘与钱猜想的一个反例及$\mathbb{Z}_p$中受限和集的改进逆定理
Infinite Families of Counterexamples to a Conjecture of Liu and Qian and a Refined Inverse Theorem for Restricted Sumsets in $\mathbb{Z}_p$
- School of Mathematics and Statistics, Liaoning University（辽宁大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对Liu和Qian关于$\mathbb{Z}_p$中受限和集临界对的猜想，在边界情形给出反例，并在$|A|+|B|\le p-1$假设下证明改进的逆定理。
AI中文摘要：
设$p$为素数，$A,B$为循环群$\mathbb{Z}_p$的非空子集，且$|A|\neq |B|$。Alon--Nathanson--Ruzsa定理给出下界\\[ |A\rplus B|\ge \min\{p,\\,|A|+|B|-2\}, \\] 其中$A\rplus B=\{a+b:a\in A,\\ b\in B,\\ a\neq b\}$为受限和集。刻画所有达到等号的临界对$(A,B)$的逆问题由Alon、Nathanson和Ruzsa于1996年提出，至今仍未解决。近期，Liu和Qian在至少一个集合为等差数列的假设下解决了该逆问题，并对一般情形提出了一个猜想。本文首先给出他们猜想的一个反例，该反例出现在边界情形$|A|+|B|=p$中。此反例表明，在没有额外限制时原猜想不成立。受此启发，我们在自然假设$|A|+|B|\le p-1$下提出并证明了一个改进的逆定理。我们的证明依赖于Liu和Qian的结果以及一个新的计数论证。
英文摘要：
Let $p$ be a prime and let $A,B$ be nonempty subsets of the cyclic group $\mathbb{Z}_p$ with $|A|\neq |B|$. The Alon--Nathanson--Ruzsa theorem gives the lower bound $|A\rplus B|\ge \min\{p,\,|A|+|B|-2\},$ where $A\rplus B=\{a+b:a\in A,\ b\in B,\ a\neq b\}$ is the restricted sumset. The inverse problem of characterizing all critical pairs $(A,B)$ attaining equality was posed by Alon, Nathanson, and Ruzsa in 1996 and remains open. Recently, Liu and Qian solved the inverse problem under the assumption that at least one of the sets is an arithmetic progression, and proposed a conjecture for the general case. The main purpose of this paper is to show that this conjecture fails in the boundary case $|A|+|B|=p$. More precisely, we construct infinite families of non-arithmetic critical pairs $(A,B)$ with $|A|+|B|=p$ and $|A\rplus B|=p-2$ for every prime $p\ge 11$. These families show that the boundary case is fundamentally different from the non-boundary case. Motivated by this, we formulate and prove a refined inverse theorem under the natural hypothesis $|A|+|B|\le p-1$, which excludes the boundary.
