素域的乘法子群不是和集
Multiplicative Subgroups of Prime Fields Are Not Sumsets
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中文总结 AI 辅助
研究素域乘法子群能否写成和集形式的问题，基于汉森 - 佩特里迪斯多项式方法及卡尔米宁的成果，通过统一论证得出结论：要么加项之一为单元素集，要么\(|A| = |B| = 2\)且\(|H| = 4\)，真乘法子群不能写成\(|A|,|B|>2\)的\(A + B\)形式。
中文摘要 AI 辅助
设\(H\leq\mathbb{F}_p^*\)是一个真乘法子群，且假设\(H = A + B\)，其中\(A,B\subseteq\mathbb{F}_p\)。我们证明要么其中一个加项是单元素集，要么\(|A| = |B| = 2\)且\(|H| = 4\)。特别地，\(\mathbb{F}_p^*\)的任何真乘法子群都不能写成\(|A|,|B|>2\)的\(A + B\)形式。我们的证明基于汉森 - 佩特里迪斯多项式方法以及卡尔米宁对二次剩余的萨尔科齐猜想的后续解决。利用卡尔米宁的\(|A| = |B|\)定理作为结构输入，我们发展了适用于任意指数乘法子群的统一组合和算术论证。
英文摘要
Let $H \leq \mathbb{F}_p^*$ be a proper multiplicative subgroup, and suppose that $H = A+B$ for some $A,B \subseteq \mathbb{F}_p$. We prove that either one of the summands is a singleton, or $|A|=|B|=2$ and $|H|=4$. In particular, no proper multiplicative subgroup of $\mathbb{F}_p^*$ can be written as $A+B$ with $|A|,|B|>2$. Our proof builds on the Hanson-Petridis polynomial method and Kalmynin's subsequent resolution of Sárközy's conjecture for quadratic residues. Using Kalmynin's $|A|=|B|$ theorem as a structural input, we develop uniform combinatorial and arithmetic arguments which apply to multiplicative subgroups of arbitrary index.
