Hong 2026 strategy proposal covering lemniscates erdos problem

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claim_3_1: Hong's unproven mean-centered partition claim: for a monic polynomial, some partition of the components of the open lemniscate, each block covered by a disk about its weighted root mean, has radius sum at most 2. The note states it as a claim and does not prove it.

corollary_4_3: In Hong's minimal-counterexample setup, distinct blocks A and B of the minimal partition, with a = n_A and b = n_B roots, have weighted root means more than min{(a+b) r(A)/a, (a+b) r(B)/b} apart.

lemma_4_1: In Hong's minimal-counterexample setup for the mean-centered partition claim, any two distinct blocks A and B of the minimal partition satisfy r(A union B) > r(A) + r(B).

lemma_4_2: In Hong's minimal-counterexample setup for the mean-centered partition claim, splitting a block S of the minimal partition into two nonempty parts U and V never lowers the radius sum: r(U) + r(V) >= r(S).

lemma_5_1: Hong's internal product bound: if z_S is a point of K_S at distance r(S) from the root mean of a block S of the minimal partition, the product of the distances from z_S to the n_S roots of S is at most (sqrt 2 r(S))^{n_S}; with |f(z_S)| = 1 this gives r(S) >= 1/(sqrt 2 |Q_S(z_S)|^{1/n_S}).

lemma_7_1: Hong's identity T_S = 1/lambda_S for every block S of the minimal partition, where T_S is the maximum of |P_S| on K_S and lambda_S the minimum of |Q_S| on K_S, with f = P_S Q_S split by the roots inside and outside S.

proposition_6_1: Hong's conditional local bound: assuming the mean-centered partition claim for monic polynomials of lower degree, every proper block S of a radius-minimizing counterexample of degree N has M_S >= 2^{-n_S}, equivalently r(S) <= 2 T_S^{1/n_S}, or rho_S <= 1.

section_8_4: Hong's statement of the open step: in the multi-block case a counterexample to the mean-centered partition claim forces the layer-cake integral of N(u,t) over 0 < u < 1 and t > 0 to exceed 1, and the note leaves proving that integral at most 1 as the remaining task.

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Boon Qing Hong, Strategy Proposal on Covering Lemniscates for Erdős Problem #509. unpublished note (2026). No notice is printed in the file, and the author-shared Google Drive file it was taken from states no terms (https://drive.google.com/file/d/1RepW4A5-6gK83j4BcVSemqfpL3JnZjxM/view); the term is unstated.

The note sets out a proposed route to Erdős problem 509, whether the set $K=\{\lvert f\rvert\le1\}$ of a monic non-constant polynomial $f$ can be covered by closed disks of total radius at most $2$, recalling that Pommerenke proved $2$ achievable when the open set $E=\{\lvert f\rvert<1\}$ is connected, with one disk of radius $2$ about the mean of the roots (p. 1). Its central Claim 3.1 (p. 2) asserts that some partition of the connected components of $E$, each block covered by the closed disk about its weighted root mean of radius $r(S)$, has radius sum at most $2$; the note states it and does not prove it. Section 4 sets up a minimal counterexample $p_*$ and records Lemma 4.1 (merge irreducibility), Lemma 4.2 (split irreducibility) and Corollary 4.3 (centroid separation), all on p. 2. Section 5 proves Lemma 5.1 (p. 3), the internal product bound $\prod_j\lvert z_S-\alpha_j\rvert\le(\sqrt2\,r(S))^{n_S}$ at an extremal point $z_S$, which with $\lvert f(z_S)\rvert=1$ gives inequality (1), $r(S)\ge1/(\sqrt2\,\lvert Q_S(z_S)\rvert^{1/n_S})$ (p. 4). Section 6 defines the local sharpness factor $\rho_S=r(S)/(2T_S^{1/n_S})$ and proves Proposition 6.1 (p. 4), $\rho_S\le1$ for proper blocks, under the hypothesis that Claim 3.1 holds in lower degree. Lemma 7.1 (p. 5) identifies $T_S$ with the reciprocal of the minimum of $\lvert Q_S\rvert$ on $K_S$, and Section 7.3 rewrites the radius sum as a layer-cake integral of a block count $N(u,t)$. Section 8.3 bounds $\rho_S$, when $L_S$ is connected, through logarithmic capacity and the Barnard--Pearce--Solynin refinement of Faber's inequality, and Section 8.4 (p. 10) leaves open the decisive step, the bound $\int_0^1\int_0^\infty N(u,t)\,dt\,du\le1$ in the multi-block case; the note says the truncation to $u<1$ is not justified for a single block. Section 9 offers Pólya's projection theorem as an input it does not complete. The note credits ChatGPT 5.5 Pro for most of its details (p. 1). It proves no case of problem 509.

Source: https://drive.google.com/file/d/1RepW4A5-6gK83j4BcVSemqfpL3JnZjxM/view.

Read status: claims checked for Claim 3.1, Lemmas 4.1, 4.2, 5.1 and 7.1, Corollary 4.3, inequality (1), Proposition 6.1 and Sections 7.3 to 9, read clause by clause on the page images of the print; the short proofs of Lemmas 4.1, 4.2, 5.1 and 7.1 and Corollary 4.3 were followed, and that of Proposition 6.1 for its structure. Claim 3.1 and the Section 8.4 target are unproven in the note. Nothing here is independently reviewed.

Bears on. #509: Claim 3.1, if true, would answer the problem's question yes with mean-centered disks; the note states it without proof, and its lemmas concern a hypothetical minimal counterexample to it. The note settles no case of the problem; it is a different document from the write-up behind the problem's Hong 2026 claim page.

Results.

• Claim 3.1 (p. 2): unproven; some admissible partition $p$ has $\sum_{S\in p}r(S)\le2$, so the disks $\overline D(c_S,r(S))$ cover $K$ with total radius at most $2$.
• Lemma 4.1 (p. 2): distinct blocks $A,B$ of $p_*$ satisfy $r(A\cup B)>r(A)+r(B)$.
• Lemma 4.2 (p. 2): a nontrivial split $S=U\sqcup V$ of a block of $p_*$ has $r(U)+r(V)\ge r(S)$.
• Corollary 4.3 (p. 2): distinct blocks of $p_*$ with $a=n_A$, $b=n_B$ have $\lvert c_A-c_B\rvert>\min\{\frac{a+b}a r(A),\frac{a+b}b r(B)\}$.
• Lemma 5.1 (p. 3) and inequality (1) (p. 4): the internal product bound $(\sqrt2\,r(S))^{n_S}$ at an extremal point, and the radius lower bound it gives.
• Proposition 6.1 (p. 4): assuming Claim 3.1 in lower degree, $\rho_S\le1$ for every proper block of a radius-minimizing counterexample.
• Lemma 7.1 (p. 5): $T_S=\lambda_S^{-1}$ for every block of $p_*$.
• Section 8.4 (p. 10): the open layer-cake target $\int_0^1\int_0^\infty N(u,t)\,dt\,du\le1$ in the multi-block case.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.