Problem 1038

Erdős Problem #1038: declared status 'open'. Formalized: yes. Determine the infimum and supremum of$$\lvert \{ x\in \mathbb{R} : \lvert f(x)\rvert < 1\}\rvert$$as $f\in \mathbb{R}[x]$ ranges over all non-constant monic polynomials, all of whose roots are real and in the interval $[-1,1]$. Current best: A problem of Erd\H{o}s, Herzog, and Piranian [EHP58], who proved that the measure of the set in question is always at most $2\sqrt{2}$ under the assumption that all the roots are in $\{-1,1\}$, and conjecture this is the best possible upper bound. They also note that the infimum of the set in question is less than $2$, as witnessed by $f(x)=(x+1)(x-1)^m$ for $m\geq 3$. They further conjectured that, if the roots are restricted to $[-2,2]$, then$$\lvert \{ x\in \mathbb{R} : \lvert f(x)\rvert < 1\}\rvert\geq n^{-c}$$for an absolute constant $c>0$. The current best known bounds (see the discussion in the comments) are$$1.519\approx 2^{4/3}-1\leq \inf \leq 1.835\cdots$$and$$\sup = 2\sqrt{2}\approx 2.828.$$ References [EHP58] Erd\H{o}s, P. Prize: no. Tags: analysis.