发表机构
Yonsei University; KAIST; Institute for Basic Science (IBS)（延世大学; 韩国科学技术院; 基础科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限阿贝尔群中$k$重和集的阈值，给出随机Cayley和超图独立数的新上界，从而改进并推广了Alon-Pham及Balogh等人的结果。
AI 中文摘要
我们记$f_k(\Gamma)$为最大的整数，使得有限阿贝尔群$\Gamma$中任意大小至少为$|\Gamma| - f_k(\Gamma)$的子集都是$k$重和集。推广Alon和Pham最近的一个结果，我们证明对所有有限阿贝尔群$\Gamma$和整数$k \geq 2$，有$f_k(\Gamma) \leq \widetilde{O} \left(n^{(2k-1)/(4k-3)}\right)$，其中$n = |\Gamma|$。此外，我们还证明若$\Gamma$没有阶整除$k$的非平凡元素，则下界$f_k(\Gamma) \geq \widetilde{\Omega} \left(n^{1/k}\right)$成立。我们的上界改进了Balogh、Liu和Sharifzadeh之前的结果，并在$k = 2$时恢复了Alon和Pham的界。证明依赖于随机Cayley和超图的独立数的一个新上界，这可能具有独立的意义。
英文摘要
We denote by $f_k(Γ)$ the largest integer with the property that every subset of a finite abelian group $Γ$ of size at least $|Γ| - f_k(Γ)$ is a $k$-fold sumset. Extending a recent result of Alon and Pham, we prove that $$ f_k(Γ) \leq \widetilde{O} \left(n^{(2k-1)/(4k-3)}\right) $$ holds for all finite abelian groups $Γ$ and integers $k \geq 2$, where $n = |Γ|$. Additionally, we also show that the lower bound $f_k(Γ) \geq \widetildeΩ \left(n^{1/k}\right)$ holds if $Γ$ has no nontrivial element of order dividing $k$. Our upper bound improves a previous result of Balogh, Liu, and Sharifzadeh, and recovers the bound of Alon and Pham in the case $k = 2$. The proof relies on a new upper bound for the independence number of random Cayley sum hypergraphs, which may be of independent interest.
Comments21 pages
