Problem Statement
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 $>(\log n)^{-O(1)}$)?
Categories:
Polynomials Analysis
Progress
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 $\ll (\log\log n)^{-1}$.
In [EHP58] they also ask to determine the polynomials which achieve the minimum possible value of this measure.
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.
Source: erdosproblems.com/116 | Last verified: January 13, 2026