Shortest paths in polynomial lemniscate sublevel sets and a problem of Erd\H{o}s

Abstract

Let $f(z)=\prod_{j=1}^{n}(z-a_j)$ be monic, with all zeros in the closed unit disk, and put $E_f=\{z\in\mathbb{C}: |z|\leq 1,\ |f(z)|\leq 1\}$. 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 $n$, $c\sqrt{\log n}\leq S(n)\leq \pi n$ with an absolute constant $c>0$. This proves the qualitative unboundedness predicted by Erd\H{o}s. The proof combines an explicit geometric maze, Green-function and Faber-polynomial estimates, analytic quantization of circle measures, and a reciprocal-sweeping upper bound.