AI 中文总结
该研究建立实直线紧子集容许最优多项式网格的位势论充分条件，结合Andrievskii估计证明一致完美紧子集存在最优多项式网格，经乘积论证得平面康托尔尘集上的最优网格。
AI 中文摘要
我们为实直线的紧子集建立了容许最优多项式网格的位势论充分条件。该准则将范数集的基数限制为多项式次数的线性项加上高度为α/n的格林次水平集的连通分支数，其中α>0为固定值。将该原理与Andrievskii给出的估计相结合，我们证明了ℝ中每个一致完美紧子集都容许最优多项式网格。随后通过标准乘积论证可在有限笛卡尔积上生成最优网格；特别地，这得到了平面康托尔尘集C×C上的最优网格，该集合是自相似、完全不连通且内部为空的。
英文摘要
We establish a potential-theoretic sufficient condition for a compact subset of the real line to admit an optimal polynomial mesh. The criterion bounds the cardinality of a norming set by a linear term in the polynomial degree plus the number of connected components of a Green sublevel at height $α/n$, where $α>0$ is fixed. By combining this principle with an estimate due to Andrievskii, we prove that every uniformly perfect compact subset of $\mathbb{R}$ admits an optimal polynomial mesh. A standard product argument then produces optimal meshes on finite Cartesian products; in particular, it yields an optimal mesh on the planar Cantor dust $C\times C$, which is self-similar, totally disconnected, and has empty interior.
Comments15 pages
