---

Statement

Notation (pp. 125--126): $f$ is a monic polynomial (1), $E$ the set where $|f|<1$, $L$ the real axis and $I=[-1,1]$; $|\cdot|$ on subsets of $L$ is linear measure.

Theorem 3 (p. 131). "If all the zeros of (1) lie at the endpoints of $I$, then $|E\cap L|\leq2\sqrt2$."

Immediately before it, the paper writes that the theorem "suggests the conjecture that if all the $x_\nu$ lie on $I$, then $|E\cap L|\leq2\sqrt2$" (p. 131); see Problem 1. The bound is attained by $x^2-1$, whose set $E\cap L$ is two intervals of length $\sqrt2$ each (p. 127).

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Theorem 3 on p. 131, its proof on pp. 131--132. The copy read is identified on the source card.

Read depth. Claims checked: the statement and the conjecture before it were read on the page image of p. 131 on 2026-10-08; the proof was read for structure, not checked. Nothing here is independently reviewed.

Proof pointer

Pages 131--132. The cases with all zeros at one endpoint, or equally many at each, are immediate, so it suffices to treat $g(x)=|x+1||x-1|^m$ with real $m>1$. Then $E(g)\cap L$ has two components; the left one is shorter than $\sqrt2-(m-1)/(m+1)$ by Theorem 1 and the monotonicity of $g$ on $I$, and the proof reduces to $g(\sqrt2+(m-1)/(m+1))>1$, settled by showing that $\log(\sqrt2-2/(m+1))^m$ increases in $m$ from its value at $m=1$.

Dependencies

Theorem 1.

Bears on

• #1038: the supremum the problem asks for, restricted to the polynomials whose zeros are all $\pm1$. The theorem gives the bound $2\sqrt2$ only in that class; the general case is the paper's conjecture, not a result.