Erdős Problem #116

Let \(p(z)=\prod_{i=1}^n (z-z_i)\) for \(\lvert z_i\rvert \leq 1\). Is it true that \[\lvert\{ z: \lvert p(z)\rvert <1\}\rvert>n^{-O(1)}\] (or perhaps even \(>(\log n)^{-O(1)}\))? Conjectured by Erdős, Herzog, and Piranian [EHP58]. The lower bound \(\gg n^{-4}\) follows from a result of Pommerenke [Po61]. The lower bound \(\gg (\log n)^{-1}\) was proved by Krishnapur, Lundberg, and Ramachandran [KLR25]. Wagner [Wa88] proves, for \(n\geq 3\), the existence of such polynomials with \[\lvert\{ z: \lvert p(z)\rvert <1\}\rvert \ll_\epsilon (\log\log n)^{-1/2+\epsilon}\] for all \(\epsilon>0\). Krishnapur, Lundberg, and Ramachandran [KLR25] improved this upper bound to \(\ll (\log\log n)^{-1}\). Pólya [Po28] showed the upper bound \(\lvert\{ z: \lvert p(z)\rvert <1\}\rvert \leq \pi\) always holds, and this is achieved only when the \(z_i\) are identical.