发表机构
Shenzhen MSU–BIT University; Guangdong Laboratory of Machine Perception and Intelligent Computing（深圳北理莫斯科大学; 广东省机器感知与智能计算实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过分离的增长最小化基和加权中位数技术，将平面点集的点积数下界改进至|P|^{199/295}，并证明固定点问题无线性下界，核心结果经Lean 4验证。
AI 中文摘要
对于每个有限的 $P\subset\mathbb{R}^2$，我们证明 $|\{p\cdot q: p,q\in P\}|\gg |P|^{199/295}$，其中常数是绝对的且没有对数损失，这里 $199/295 = 2/3+7/885$。这改进了 Kokkinos (arXiv:2502.12727) 的界 $2/3+7/1425$，而后者又改进了 Hanson、Roche-Newton 和 Senger 的第一个超阈值界。主要的新工具是不等式 $|F^{(2)}G^{(2)}/(F^{(2)}G)| \le |AB|(|AF|/|A|)^4(|BG|/|B|)^3$：每个径向轮廓保留其自身的 Petridis 增长最小化子集，且两个子集的乘积作为两个增长算子的公共基；一个素数幂构造表明在该假设下这是紧的。第二个工具是 Roche-Newton 和 Wong 的挤压论证中二进选择的加权中位数替代，将其七因子展开器升级为无对数形式；附录给出了基于 Solymosi-Zahl 关联定理的完整证明。我们用近极值构型的第三矩结构定理补充下界，并给出一个构造表明固定点问题没有线性下界：$\max_{p\in P}|p\cdot P| = O(|P|/\sqrt{\log |P|})$ 是可达到的。代数核心、中位数引理、曲面恒等式和固定点构造已在 Lean 4 中正式验证并作为附件文件包含。
英文摘要
For every finite $P\subset\mathbb{R}^2$ we prove $|\{p\cdot q: p,q\in P\}|\gg |P|^{199/295}$, with an absolute constant and no logarithmic loss, where $199/295 = 2/3+7/885$. This improves the bound $2/3+7/1425$ of Kokkinos (arXiv:2502.12727), which in turn had improved the first superthreshold bound of Hanson, Roche-Newton, and Senger. The main new ingredient is the inequality $|F^{(2)}G^{(2)}/(F^{(2)}G)| \le |AB|(|AF|/|A|)^4(|BG|/|B|)^3$: each radial profile retains its own Petridis growth-minimizing subset, and the product of the two subsets serves as a common base for both growth operators; a prime-power construction shows this is sharp under its hypotheses. The second ingredient is a weighted-median replacement for the dyadic selection in the squeezing argument of Roche-Newton and Wong, upgrading their seven-factor expander to a logarithm-free form; an appendix gives the complete proof from the Solymosi-Zahl incidence theorem. We complement the lower bound with a third-moment structure theorem for near-extremal configurations and a construction showing that the pinned problem admits no linear lower bound: $\max_{p\in P}|p\cdot P| = O(|P|/\sqrt{\log |P|})$ is attainable. The algebraic core, the median lemma, the surface identities, and the pinned construction are formally verified in Lean 4 and included as ancillary files.
