极值情形下平移乘法子群的乘法不可约性
Multiplicative irreducibility of shifted multiplicative subgroups in the extremal case
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中文总结 AI 辅助
研究素域中平移乘法子群乘法不可约性问题，在\(\lambda\in\mathbb{F}_p^*\setminus G\)时，借助素域中的斯捷潘诺夫界，于等式情形完全解决该问题，此前金、叶和柳已有相关研究基础。
中文摘要 AI 辅助
最近，卡尔米宁证明了关于素域中二次剩余集加法不可约性的萨尔科齐猜想。最近，金、叶和柳开始研究该猜想的平移乘法子群的乘法类似物。具体而言，他们表明对于奇素数\(p\)，\(\mathbb{F}_p^*\)的一个真乘法子群\(G\)以及\(\lambda\in G\)，不存在\(|A|,|B|\geq2\)的集合\(A,B\subseteq\mathbb{F}_p^*\)使得\(AB=(G - \lambda)\setminus\{0\}\)。本文中，当\(\lambda\in\mathbb{F}_p^*\setminus G\)时，我们从素域中的斯捷潘诺夫界在等式情形下完全解决了这个问题。
英文摘要
In a recent breakthrough, Kalmynin proved a conjecture of Sárközy on additive irreducibility of the set of quadratic residues in a prime field. More recently, Kim, Yip, and Yoo initiated the study of a multiplicative analogue of the conjecture for shifted multiplicative subgroups. Specifically, they showed that for an odd prime $p$, a proper multiplicative subgroup $G$ of $\mathbb F_p^*$, and $λ\in G$, there do not exist sets $A,B\subseteq \mathbb F_p^*$ with $|A|,|B|\ge 2$ such that $AB=(G-λ)\setminus\{0\}$. In this paper, when $λ\in \mathbb F_p^* \setminus G$, we completely resolve this problem in the equality case from a Stepanov bound in a prime field.
