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Statement

$f(z)=\prod_{\nu=1}^n(z-z_\nu)=z^n+\cdots$ and $E=\{|f(z)|\le1\}$, the lemniscate domain (p. 97). Problem 8 of Erdős, Herzog and Piranian asks whether $\sum\max(0,d_j-1)$ over the diameters $d_j$ of the components of $E$ is bounded over all monic polynomials, and Problem 9, in its revised form, whether the number of components of diameter greater than a fixed $l>1$ is bounded (pp. 97--98). Quoted (p. 98): "The following theorem shows that the answer to these two questions is negative, even with $d_j-l$ ($l<4$) instead of $d_j-1$ in Problem 8, and with any $l<4$ in Problem 9."

Theorem 1 (p. 98). "For each $0<l<4$ and $k=1,2,\cdots$, one can find a polynomial $f(z)=z^n+\cdots$ such that $E=\{|f(z)|\le1\}$ has at least $k$ different components of diameter greater than or equal to $l$."

The bound $l<4$ cannot be relaxed: by Theorem 6 (p. 102), which sharpens Pólya's bound recalled there, the projection of $E$ onto any line has measure at most $4\cdot2^{-1/n}<4$, so no component has diameter $4$ or more.

Source. Ch. Pommerenke, On metric properties of complex polynomials, Michigan Math. J. 8 (1961), no. 2, 97--115; Theorem 1 with its proof on printed p. 98 (PDF p. 2 of the publisher's scan), the approximation theorem on p. 97 (PDF p. 1), read on the page images (the scan has no text layer). The copy read is identified in the source digest.

Read depth. Claims checked: the statement, the two problems as the paper recalls them and the approximation theorem were read clause by clause on the page images on 2026-09-22. The proof (one paragraph) was read in full and its steps followed, with the observation recorded below. Nothing here is independently reviewed.

Proof pointer

Page 98. The construction stacks $k$ horizontal segments of length $l$, $[i\mu\delta,\,l+i\mu\delta]$ for $\mu=1,\ldots,k$, a distance $\delta>0$ apart, and takes their union as $F$. Each segment has capacity $l/4<1$, and a continuity argument keeps $\operatorname{cap}F$ below 1 once $\delta$ is small. The approximation theorem (p. 97, from the paper's [5]: for a closed bounded $F$ with $\operatorname{cap}F=1$ and any $\varepsilon,\eta>0$ there are $\rho\in(1,1+\eta)$ and a monic $f$ whose lemniscate curve $\{|f|=\rho^n\}$ contains $F$ in its interior and is contained in an $\varepsilon$-neighborhood of $F$) then supplies a polynomial whose set $E$ contains $F$ and lies within distance $\delta/3$ of it. Because the segments are $\delta$ apart, no component of $E$ meets two of them, so the $k$ segments lie in $k$ distinct components of $E$, each of diameter at least $l$.

A filing observation, not a review verdict: the theorem is quoted for $\operatorname{cap}F=1$ and the level $\rho^n>1$, and the proof applies it to a set of capacity below 1 at level 1 without spelling out the rescaling that absorbs $\rho$; the step was not reconstructed here. A second observation: since $F$ lies in the interior of the lemniscate set, the segments lie in the open set $\{|f|<1\}$, which is that interior, so the $k$ components of the open set containing them also have diameter at least $l$.

Dependencies

Within the paper: the approximation theorem of p. 97, which the paper cites to Fekete, Über den transfiniten Durchmesser ebener Punktmengen III, Math. Z. 37 (1933), 635--646 (the paper's [5]; not held), and the capacity of a segment. Nothing else.

Bears on

• Problem 511: the negative answer to the question whether $\{|f|<1\}$ has $O_c(1)$ components of diameter above $c>1$ independently of the degree; for every $c<4$ the count is unbounded. The site's status "disproved" rests on this theorem, and the 2025 rediscovery Huang 2025 says the problem had been solved by the author.