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Statement

Setting (pp. 1--2). $f\in\mathbb{C}[z]$ is monic and non-constant, $E=\{z\in\mathbb{C}:\lvert f(z)\rvert<1\}$ and $K=\{z\in\mathbb{C}:\lvert f(z)\rvert\le1\}$. The connected components of $E$ are $E_1,\ldots,E_m$, and $K_j=\overline{E_j}$. The note calls every partition of $\{E_1,\ldots,E_m\}$ an admissible partition. For a block $S$ of one, $K_S$ is the union of the $K_j$ with $E_j\in S$; $n_S$ is the number of roots of $f$ in the components belonging to $S$, counted with multiplicity; $c_S=\frac1{n_S}\sum_{\alpha\in S}m(\alpha)\alpha$ is the weighted root mean; and $r(S)=\sup_{z\in K_S}\lvert z-c_S\rvert$.

Claim 3.1 (Mean-centered partition claim, p. 2). There is an admissible partition $p$ with $\sum_{S\in p}r(S)\le2$; consequently the closed disks $\overline D(c_S,r(S))$, $S\in p$, cover $K$ and have total radius at most $2$.

Remark 3.2 (p. 2) observes that the covering follows from the definition of $r(S)$, so the content of the claim is the radius-sum bound.

Status in the note

The note does not prove Claim 3.1. Sections 4 to 9 (pp. 2--10) set up a minimal counterexample to it and record partial estimates, and Section 8.4 (p. 10) names the step that remains open; see Section 8.4. Proposition 6.1 (p. 4) uses Claim 3.1 for lower degrees as an induction hypothesis.

For context the note recalls (p. 1) that Pommerenke proved the constant $2$ achievable when $E$ is connected, with $K$ inside the disk of radius $2$ about the mean of all the roots; that is the case $m=1$ of the claim, with the one-block partition.

Read depth

Claims checked: the definitions, Claim 3.1 and Remark 3.2 were read clause by clause on the page images of the print. There is no proof to check. Nothing here is independently reviewed.

Source. Boon Qing Hong, Strategy Proposal on Covering Lemniscates for Erdős Problem #509, unpublished note (2026), 11 pp.; the edition read is named on the source card. The note credits ChatGPT 5.5 Pro for most of its details (p. 1).

Bears on

• Problem 509: Claim 3.1, if true for every monic non-constant $f$, would answer the problem's question yes with disks of a special form, centered at weighted root means of blocks of components. The note states it without proof, so it settles no case of the problem.