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Subhajit Ghosh and Koushik Ramachandran, A note on the Erdős minimal area problem, Proc. Amer. Math. Soc., published online 28 August 2026, DOI 10.1090/proc/17897 (arXiv:2604.03036, v1 of 3 April 2026, v3 of 27 August 2026), answer the first question negatively and settle the second for compact sets of capacity above $1$. With $A_n(K)$ the minimal area of $\{z:|p(z)|<1\}$ over monic $p$ of degree $n$ with all zeros in $K$ and $\vartheta(K)=\inf_nA_n(K)$, Theorem 3.1 states that a compact $K$ with $\operatorname{cap}(K)=t>1$ has $\limsup_n\frac1n\log A_n(K)\le-2\log t$, so $\vartheta(K)=0$, the Fekete polynomials of $K$ giving the upper bound; Theorem 3.3 gives the matching lower bound, so that $\frac1n\log A_n(K)\to-2\log t$ for every such $K$ with no regularity assumption, and the Fekete polynomials are asymptotic minimizers. Example 2.1 answers the first question negatively at every capacity $t\in(0,1)$: two short symmetric intervals $[j,j+\varepsilon]\cup[-j-\varepsilon,-j]$ of capacity $t$ have $\vartheta$ below any prescribed bound once $j$ is large, while the disc of radius $t$ has $\vartheta\ge\pi r(t)^2$ by the theorem of Erdős and Netanyahu. A final section proves $A_n([-2,2])\le c_1/n$ (Theorem 4.1; Remark 4.2 announces a matching lower bound without proof) and sets it against the lower bound $A_n(\overline{\mathbb D})\ge c/\log n$ of Krishnapur, Lundberg and Ramachandran, so the decay rate at capacity $1$ is not universal. The introduction states that the general compact case of capacity $1$ remains open and cites the smooth case of Krishnapur, Lundberg and Ramachandran. The statements above are those of arXiv v3; the proofs are not checked here.

Covers. The first question, answered negatively at every capacity in $(0,1)$ by Example 2.1, beside Aletheia's capacity-zero pair; this is the part the page's value records. It also settles the second question for compact sets of capacity above $1$, $\mu(K)=0$ with exponential decay of the degree-$n$ minimal area at the sharp rate, a partial yes to that question which the page's value does not record. Its Theorem 4.1 also gives $\mu([-2,2])=0$, a capacity-one case that Erdős, Herzog and Piranian had noted; it says nothing about other compact sets of capacity exactly $1$ beyond the cited smooth case of Krishnapur, Lundberg and Ramachandran, nor about unbounded closed sets.

Refereed. Proceedings of the American Mathematical Society, DOI 10.1090/proc/17897, published online 28 August 2026 according to the Crossref record (checked 2026-10-07; no volume or pages assigned); the arXiv comments of v3 state the acceptance. The site's commentary, last edited 1 February 2026, does not name the paper; a thread comment of 6 April 2026 pointed to the preprint, and later claimants on the second question cite the theorem for the case of capacity above $1$.