---

Statement

Notation (pp. 125--126): $f$ is a monic polynomial (1) with zeros $x_\nu$, $E$ the set where $|f|<1$, $L$ the real axis, $I=[-1,1]$ and $I_r=[-r,r]$; $|E\cap L|$ is linear measure.

Problem 1 (p. 131). "To determine the supremum and the infimum of the quantity $|E\cap L|$, under the hypothesis that the $x_\nu$ lie on the interval $I_r$ (if no restriction is placed on the $x_\nu$, then the supremum is 4; see [6, p. 229]). The following theorem suggests the conjecture that if all the $x_\nu$ lie on $I$, then $|E\cap L|\leq2\sqrt2$."

The "following theorem" is Theorem 3, the case of zeros at $\pm1$. After its proof the paper adds (p. 132) that for zeros on $I$ the infimum of $|E\cap L|$ is less than $2$, by the example $(x+1)(x-1)^m$ with $m\ge3$, and that "Careful computations show that the infimum can not be approached by polynomials of the form $(x-1)^k(x+1)^m$." For zeros on $I_r$ it notes (p. 132) that the infimum is $0$ when $r\ge2$, that the minimum for fixed $n$ is $O((2/r)^n)$ when $r>2$, and for $r=2$ it conjectures $|E\cap L|>n^{-c}$ (no quantifier on $c$ is printed).

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Problem 1 on p. 131, the remarks on p. 132. The copy read is identified on the source card.

Read depth. Claims checked: the problem and the remarks were read clause by clause on the page images of pp. 131--132 on 2026-10-08. Nothing here is independently reviewed.

Dependencies

Theorem 3 (the evidence for the conjecture). Bounds drawn here from the paper's theorems are on the pages of Theorem 1 (infimum at least $\sqrt2$) and Theorem 2 (supremum at most $3$).

Bears on

• #1038: the problem's question is Problem 1 in the case $r=1$, the supremum and infimum of $|E\cap L|$ over monic polynomials with all zeros in $[-1,1]$; the paper conjectures $2\sqrt2$ for the supremum and states that the infimum is below $2$.