Let $F\subset \mathbb{C}$ be a closed set of transfinite diameter $1$ which is not contained in any closed disc of radius $1$. If $f(z)={\prod }_{i=1}^n(z-z_i)\in \mathbb{C}[x]$ with all $z_i\in F$ then can$$\{z:|f(z)|<1\}$$have $n$ connected components?If the transfinite diameter of $F$ is $<1$ then must this set only have at most $(1-c)n$ connected components, where $c>0$ depends only on $F$ (or just the transfinite diameter of $F$)?