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Source. T. Feng, T. Trinh, G. Bingham et al., Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems, arXiv:2601.22401v3 (5 February 2026); Section 3.2, the problem and Remark 3.2 on p. 18, the solution on pp. 18--19 with its footnote on p. 19. The result is unnumbered. The artifact is identified on the source card.

Read depth. Claims checked: the assertion, both constructions and their estimates (pp. 18--19) were read in full on the print. Nothing here is independently reviewed. A preprint.

Statement

For a closed infinite $F\subseteq\mathbb C$, $\mu(F)$ is the infimum of the area of $\{z:|f(z)|<1\}$ over the polynomials $f=\prod(z-z_i)$ with all $z_i\in F$ (p. 18). The paper proves that $\mu(F)$ is not determined by the transfinite diameter $d_\infty(F)$, with the sets

$$F_1=\{0\}\cup\{1/n:n\ge1\},\qquad F_2=\{0,R\}\cup\{1/n:n\ge1\}\cup\{R+1/n:n\ge1\}\quad(R>4).$$

Both are countable compact sets, so $d_\infty(F_1)=d_\infty(F_2)=0$; $\mu(F_1)\ge\pi/4$, while $\mu(F_2)\le2\pi/(R^2-4)$, which is below $\pi/4$ for $R$ large enough (pp. 18--19).

Proof pointer

For $F_1\subset[0,1]$, every such polynomial has modulus below $1$ on the disc $|z-1/2|<1/2$, of area $\pi/4$. For $F_2$, the polynomial $z(z-R)$ has roots in $F_2$, and a change of variables bounds the area of its lemniscate $\{|z(z-R)|<1\}$ by $2\pi/(R^2-4)$. That countable compact sets have transfinite diameter zero is used without proof in the model output; a human footnote (p. 19) says it essentially follows from Ransford's Corollary 3.2.5 and the equivalence of transfinite diameter and logarithmic capacity, and that it is easy to verify directly for the two sets.

Dependencies

Ransford, Potential Theory in the Complex Plane (1995), Corollary 3.2.5, for the capacity-zero claim (cited in the footnote, not held).

Bears on

• Problem 1040: answers its first question, whether $\mu(F)$ is determined by the transfinite diameter, in the negative. The second question, whether $\mu(F)=0$ whenever the transfinite diameter is at least $1$, is not addressed: Remark 3.2 (p. 18) says the model's argument for it was an incorrect reduction to the literature and was omitted.