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Statement

Setting (pp. 3--4). The notation of Lemma 5.1: for a block $S$, $P_S(z)=\prod_{\alpha\in S}(z-\alpha)^{m(\alpha)}$ is the monic polynomial of degree $n_S$ whose roots are those of $f$ in $S$. The note sets $T_S=\sup_{z\in K_S}\lvert P_S(z)\rvert$, $M_S=T_S/r(S)^{n_S}$ and the local sharpness factor $\rho_S=r(S)/(2T_S^{1/n_S})$.

Proposition 6.1 (Inductive local bound for proper blocks, p. 4). Assume Claim 3.1 holds for monic polynomials of degree strictly smaller than $N=\deg f$. Let $p_*$ be a radius-minimizing counterexample of degree $N$, and let $S\in p_*$ be a proper block, so $n_S<N$. Then $M_S\ge2^{-n_S}$; equivalently $r(S)\le2T_S^{1/n_S}$, or $\rho_S\le1$.

The result is conditional: its hypothesis is the claim the note sets out to prove, in lower degrees.

Proof pointer

Pp. 4--5. If $r(S)>2T_S^{1/n_S}$, apply the hypothesis to $P_S$ at level $T_S$ (through the rescaled monic polynomial $w\mapsto T_S^{-1}P_S(T_S^{1/n_S}w)$; the print calls the claim "Claim 2.1" [sic] at this point). The lemniscate $\{\lvert P_S\rvert\le T_S\}$ contains $K_S$, each component of $S$ lies in one component of $\{\lvert P_S\rvert<T_S\}$, and the resulting partition refines $S$ into blocks with the same mean centers and total radius at most $2T_S^{1/n_S}<r(S)$, against the minimality of $p_*$.

Read depth

Claims checked: the definitions and Proposition 6.1 were read clause by clause on the page images of the print, and the proof on pp. 4--5 was followed for its structure. Nothing here is independently reviewed.

Dependencies

• Claim 3.1 in lower degree, as a hypothesis.

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 Boon Qing Hong, Strategy Proposal on Covering Lemniscates for Erdős Problem #509, unpublished note (2026), 11 pp.; the edition read is named on the source card.

Bears on

• Problem 509: a step of an induction on the degree toward Claim 3.1, conditional on that claim in lower degrees; it settles no case of the problem.