On Hilbert lemniscate theorem for a system of continua

TL;DR

This paper investigates how well finite unions of continua in the complex plane can be approximated from outside by lemniscates, using Green function level lines to measure the approximation rate.

Contribution

It introduces a method to analyze the approximation rate of complex continua by lemniscates through Green function level lines.

Findings

Established a relationship between lemniscate approximation and Green function level lines.

Provided estimates for the rate of approximation of continua by lemniscates.

Extended classical lemniscate theorems to systems of continua.

Abstract

Let $K$ be a compact set in the complex plane consisting of a finite number of continua. We study the rate of approximation of $K$ from the outside by lemniscates in terms of level lines of the Green function for the complement of $K$.