发表机构
Kansas State University; Hong Kong University of Science and Technology; Institute for Basic Science（堪萨斯州立大学; 香港科技大学; 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
本文针对素域真乘法子群的加性分解分类问题，在 Kalmynin 等人的通用框架基础上，给出了一种简化论证的全新自包含证明，明确唯一非平凡例子是阶为4的子群。
AI 中文摘要
Sárközy 猜想，足够大素数的非零二次剩余不存在非平凡加性分解。Hanson 和 Petridis 证明了几乎所有素数满足该猜想，Kalmynin 完成了完整证明，还开发了乘法子群加性分解的通用框架。近期 Rudnev 和 Tyrrell 用该框架对素域真乘法子群的所有加性分解分类，表明唯一非平凡例子是阶为4的子群。我们给出该分类的全新自包含证明，简化了 Kalmynin 及 Rudnev、Tyrrell 的论证。
英文摘要
We develop a local-to-global differential framework for additive decomposition problems involving multiplicative subgroups of prime fields. Starting from Hanson--Petridis-type auxiliary polynomials, we use degree bounds, in the spirit of Stepanov's method, to lift local coefficient relations at their roots to global differential identities. This yields a unified treatment of \[ H=A+B,\qquad H\cup\{0\}=A-A,\qquad H=A\mathbin{\widehat{+}} A,\qquad H\cup\{0\}=A\mathbin{\widehat{+}} A, \] where $H$ is a proper multiplicative subgroup. This circle of problems is motivated by Sárközy's conjecture on the additive irreducibility of nonzero quadratic residues and its generalizations to multiplicative subgroups. Rudnev and Tyrrell recently classified all decompositions $H=A+B$, building on the approach introduced by Hanson--Petridis and further developed by Kalmynin. Our framework gives a new polynomial proof of the Rudnev--Tyrrell classification and substantially streamlines the existing proofs: it gives an independent proof of Kalmynin's equal-size theorem and reduces the classification to direct coefficient comparisons, avoiding the residue-theoretic input and more elaborate arithmetic analysis of earlier proofs. It also yields a streamlined proof of Kalmynin's resolution of a conjecture of Lev and Sonn on $H\cup\{0\}=A-A$. For the two restricted-sumset problems, we obtain complete classifications, substantially improving earlier results of Shkredov and Yip. We also establish some stability refinements.
Comments34 pages, author added, substantially revised version incorporating arXiv:2607.25711, with substantially strengthened results
