The Capacity of some Sets in the Complex Plane

Abstract

For a positive integer $s$ and for $0\le a< b,$ let $$K=K^s_{a,b}=\bigcup_{k=0}^{s-1}e^{2\pi i\frac{k}{s}}[a,b].$$ We find that the {\it capacity} $\mathrm{Cap} (K)$ of $K$ is $$\mathrm{Cap} (K) = \sqrt [s]{\frac{b^s-a^s}4}\cdot\eqno (1)$$ \par From this relation we derive several classical results, due to Akhiezer, Henrici, and Bartolomeo and He, on capacities of some sets in the complex plane. \par An extension relation (1) to more general sets in the complex plane, together with potential theoretic techniques, is then used to obtain {\it saturation} theorems pertaining to approximation by polynomials with integer coefficients.