Optimal polynomial meshes beyond geometric boundary regularity via Green sublevels

Abstract

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 $\alpha /n$, where $\alpha >0$ 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.

15 pages