Number of components of polynomial lemniscates: a problem of Erdös, Herzog, and Piranian

Let $K\subset \mathbb{C}$ be a compact set in the plane whose logarithmic capacity $c(K)$ is strictly positive. Let ${\mathcal{P}}_n(K)$ be the space of monic polynomials of degree $n,$ \emph{all} of whose zeros lie in $K\mathrm{.}$ For $p\in {\mathcal{P}}_n(K),$ its filled \emph{unit leminscate} is defined by ${\mathrm{\Lambda }}_p=\{z:\mathrm{\mid }p(z)\mathrm{\mid }<1\}\mathrm{.}$ Let $\mathcal{C}({\mathrm{\Lambda }}_p)$ denote the number of connected components of the open set ${\mathrm{\Lambda }}_p,$ and define ${\mathcal{C}}_n(K)={\max }_{p\in {\mathcal{P}}_n(K)}\mathcal{C}({\mathrm{\Lambda }}_p)\mathrm{.}$ In this paper we show that the quantity $$M(K)=\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}{\displaystyle \frac{{\mathcal{C}}_n(K)}{n}},$$ satisfies $M(K)<1$ when the logarithmic capacity $c(K)<1,$ and $M(K)=1$ when $c(K)\ge 1.$ In particular, this answers a question of Erdös et. al. posed in $1958$. In addition, we show that for nice enough compact sets whose capacity is strictly bigger than $\frac{1}{2}$, the quantity $m(K)={\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}{\displaystyle \frac{{\mathcal{C}}_n(K)}{n}}>0.$