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Ioannis Tzachristas, A solution to the Erdős Problem #1040, arXiv:2609.06050 (v1 of 5 September 2026), claims the second question in full. Writing $\vartheta(F)$ for the infimum, over all degrees $n\ge1$ and all $z_1,\ldots,z_n\in F$ with repetitions allowed, of the area of $\{z:|\prod_j(z-z_j)|<1\}$, Theorem 1.1 states that a compact $K$ with $\operatorname{cap}(K)=1$ has $\vartheta(K)=0$: for every $a>0$ there are $n$ and $z_1,\ldots,z_n\in K$ whose monic product has sublevel area below $a$, with no regularity assumption on $K$. The proof builds a centered harmonic polynomial positive outside a set of arbitrarily small area in the polynomial hull of $K$ by a Cauchy-transform singularity and a Hahn-Banach argument, realizes it as the logarithmic potential of a signed measure with bounded density against the equilibrium measure by averaging exterior harmonic measures, perturbs the equilibrium measure into a positive probability measure, and discretizes by equally weighted point masses on $K$ using convergence of potentials in planar $L^1$. Section 7 completes the question with the paper's own arguments. Proposition 7.1 recovers the vanishing of $\vartheta$ for a compact set of capacity above $1$ from the paper's Proposition 5.2, since the equilibrium potential is then positive outside a set of area zero; the paper cites the sharp exponential rate of Ghosh and Ramachandran as the stronger result, which it does not reprove. Proposition 7.2 handles unbounded $F$ by two roots far apart, without assuming a compact subset of capacity above $1$. Corollary 7.3 then states that every closed infinite $F\subset\mathbb C$ of transfinite diameter at least $1$ has $\vartheta(F)=0$: a bounded $F$ is compact and falls under Theorem 1.1 or Proposition 7.1, and an unbounded one under Proposition 7.2. The construction is existential and gives no degree bound or rate. The paper is licensed CC BY 4.0. The statements above are those of the arXiv version; the proofs are not checked here.

Submission note. Posted to erdosproblems.com as a proof claim by Ioannis Tzachristas (account Ioannis_Tzachristas) on 9 September 2026, giving "GPT-Astra" as the AI used:

My preprint addresses the remaining capacity-1 case of Erdos Problem #1040. The main idea is to construct a centered harmonic polynomial that is positive on all but a set of arbitrarily small area in the polynomial hull of K. This is represented as the logarithmic potential of a signed measure controlled by the equilibrium measure, so that a small perturbation remains a positive probability measure. Approximating this measure by equally weighted point masses on K then gives monic polynomials whose unit lemniscates have arbitrarily small area. Hence theta(K) = 0 when cap(K) = 1, completing the vanishing statement together with the known cap(K) > 1 case. The work was developed with substantial AI assistance, as disclosed in the manuscript. Priority note: this preprint was first publicly posted on 5 September 2026, before the subsequently announced proof claims for the remaining capacity-1 case. Notes: Priority note: this preprint was first publicly posted on 5 September 2026, before the subsequently announced proof claims for the remaining capacity-1 case.

Covers. The second question entirely: $\mu(F)=0$ for every closed infinite set of transfinite diameter at least $1$, the compact case of capacity exactly $1$ being the paper's own theorem. The claim says nothing about the first question, which the paper notes has the negative answer recorded on Aletheia's page and on [[problems/analysis/E1040/claims/2026_04_03_ghosh_ramachandran|the page of Ghosh and Ramachandran]].

The claim rests on no other page. The paper cites [[problems/analysis/E1040/claims/2026_04_03_ghosh_ramachandran|Ghosh and Ramachandran's theorem]] as the comparison for Proposition 7.1 and the smooth case of Krishnapur, Lundberg and Ramachandran as the model of its strategy, and consumes neither.

Standing. The result was filed on the site's proof-claims tab on 9 September 2026; the tab names GPT-Astra, and the paper's closing section says that the proposed result was obtained through the author's prompting of OpenAI's GPT-6-Astra model through Codex, with proof exploration, literature checks, drafting and internal reviews by parallel reasoning agents, that these checks are not independent human verification, and that no proof-assistant formalization or external endorsement is asserted. The claim's notes assert priority over the two claims filed on 6 September 2026, the preprint having been posted on 5 September. The thread carries no comment on the claim, the site labels the problem OPEN, no reviewer is named and nothing is refereed, so the claim stays claimed. The two Lean-backed claims of the same statement are shlummi's page and Gessel's page.