---

Statement

Notation (p. 125): $f$ is a monic polynomial (1) and $E(f)$ the set where $|f|<1$; $|E(f)|$ is area.

Problem 4 (p. 135). "Let $F$ be a closed infinite point set, and let $\mu(F)$ denote the infimum of $|E(f)|$, for polynomials (1) with all zeros in $F$. Is $\mu(F)$ determined by the transfinite diameter of $F$? In particular, is $\mu(F)$ equal to 0 whenever the transfinite diameter of $F$ is greater than or equal to 1?"

The paper continues (pp. 135--136): "For the case where $F$ is a line segment or a circular disk, the result follows from the consideration of Tchebycheff polynomials in conjunction with Theorem 8, and from Theorem 4, respectively."

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Problem 4 on pp. 135--136. The copy read is identified on the source card.

Read depth. Claims checked: the problem and the sentence after it were read clause by clause on the page images of pp. 135--136 on 2026-10-08. Nothing here is independently reviewed.

Dependencies

Theorem 4 and Theorem 8, which the paper names for the disk and segment cases; Theorem 6, which gives $\mu(F)>0$ when the transfinite diameter is below $1$ (the step is drawn on that page).

Bears on

• #1040: the problem's two questions are those of Problem 4, in the same terms.