Shortest paths in polynomial lemniscate sublevel sets and a problem of Erdős

Abstract

Let $f(z)={\prod }_{j=1}^n(z-a_j)$ be monic, with all zeros in the closed unit disk, and put $E_f=\{z\in \mathbb{C}:|z|\le 1,|f(z)|\le 1\}$. Let $S(n)$ be the largest possible shortest length of a path in $E_f$ joining $0$ to $\partial \mathbb{D}$, where the maximum is taken over all such polynomials of degree $n$. We prove that, for all sufficiently large $n$, $c\sqrt{\log n}\le S(n)\le \pi n$ with an absolute constant $c>0$. This proves the qualitative unboundedness predicted by Erdős. The proof combines an explicit geometric maze, Green-function and Faber-polynomial estimates, analytic quantization of circle measures, and a reciprocal-sweeping upper bound.