---

Statement

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

Theorem 6 (p. 135). "Let $F$ be a closed set of transfinite diameter less than 1. Then there exists a positive number $\rho(F)$ such that, for every polynomial (1) whose zeros lie in $F$, the set $E(f)$ contains a disk of radius $\rho(F)$."

The radius does not depend on the degree. The proof shows more: the disk can be taken to be one of finitely many disks $H_j$ of radius $\rho(F)$, centered at points of $F$ fixed in advance.

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

Read depth. Claims checked: the statement was read on the page image of p. 135 on 2026-10-08, and the proof was read and its steps followed, not independently checked. Nothing here is independently reviewed.

Proof pointer

Page 135. Because the transfinite diameter of $F$ is below $1$, Fekete's theory (the paper's [4], Sections 2 and 3) gives points $t_1,\dots,t_m$ of $F$ with $|\prod_j(z-t_j)|<1$ on $F$, and by continuity some $\rho>0$ keeps $\prod_j|z-s_j|<1$ on $F$ whenever each $s_j$ lies in the disk $H_j$ of radius $\rho$ about $t_j$. For $f$ with zeros in $F$, pick $s_j$ on the boundary of $H_j$ where $|f|$ is largest on $\bar H_j$. The product of the $f(s_j)$ equals, up to sign, the product over the zeros $z_\nu$ of $\prod_j(z_\nu-s_j)$, which has modulus at most $1$; so some $|f(s_j)|\le1$, and then $|f|<1$ throughout $H_j$.

Dependencies

Fekete's results on transfinite diameter (the paper's [4]); nothing else in the paper.

Bears on

• #1040: the theorem gives $|E(f)|\ge\pi\rho(F)^2$ for every $f$ with zeros in $F$, so $\mu(F)>0$ whenever $F$ is closed and infinite with transfinite diameter below $1$; this consequence is drawn here, not stated in the paper. Problem 4 asks about the complementary case, transfinite diameter at least $1$.
• #1039: the closed unit disk has transfinite diameter $1$, so the theorem does not apply to the class of Problem 3 and gives no lower bound for its inradius $\rho_n$.