AI 中文总结
该研究针对用有限齐次空间逼近球面的一致格罗莫夫-豪斯多夫间隙问题，通过ChatGPT整合多类理论结果得到定量下界，为解决核心开放问题提供了关键进展。
AI 中文摘要
设$S^n$为带固有角度量的单位圆球面，归一化后直径$\text{diam}S^n=\text{π}$。对有限齐次度量空间$X$，记$\text{δ}_n=\text{inf}_X d_{GH}(X,S^n)$。核心开放问题是：当$n\text{≥}2$时，$\text{inf}\text{δ}_n$是否大于0？Gelander定理给出固定维度下$\text{δ}_n>0$，但非一致；抽象交叉多面体构造给出通用上界$\text{δ}_n\text{≤}\text{π}/4$。ChatGPT结合小格罗莫夫-豪斯多夫误差到球面上近似有限作用的推导、Cuesta提出的近似保内积映射的对数稳定性、近表示的算子范数稳定性，以及有限传递集的Green宽度定理，对所有足够大的$n$得到定量下界$\text{δ}_n\text{≥}\frac{c}{(1+\text{log}(n+1))^2}$。
英文摘要
Let $S^n$ be the unit round sphere with its intrinsic angular metric, normalized so that $\operatorname{diam}S^n=π$. For finite homogeneous metric spaces $X$, put \[ δ_n=\inf_X d_{GH}(X,S^n). \] The main open problem is whether $\inf_{n\ge2}δ_n>0$. Gelander's theorem gives $δ_n>0$ in each fixed dimension, but not uniformly. An abstract cross-polytope construction gives the universal upper bound $δ_n\leπ/4$. In the opposite direction, ChatGPT combines the passage from small Gromov--Hausdorff error to an approximate finite action on the sphere, logarithmic stability of approximate inner-product-preserving maps due to Cuesta, operator-norm stability of almost representations, and Green's width theorem for finite transitive sets. This gives the quantitative bound \[ δ_n\ge \frac{c}{(1+\log(n+1))^2} \] for all sufficiently large $n$.
