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Whether transfinite diameter determines polynomial sublevel area

Does the transfinite diameter of a compact subset F of the complex plane determine the polynomial sublevel-area quantity μ(F)? A separate question asks whether μ(F)=0 whenever that diameter is at least one.

Why it matters and what counts as progress

Why it matters: The first implication tests whether one potential-theoretic size invariant determines a different polynomial-area invariant.

What counts as progress: Two compact sets with identical transfinite diameter but different μ-values, or a determination theorem; this does not by itself answer the separate diameter-at-least-one question.

How the problem developed

  1. 29 Jan 2026 · Counterexample [1][2]

    Tony Feng and collaborators, with Aletheia (Gemini Deep Think) Reported that μ(F) is not determined by transfinite diameter: two countable compact sets both have diameter zero but yield different bounds for the polynomial sublevel area.

AI contribution

Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems

Tony Feng, Trieu Trinh, Garrett Bingham, Jiwon Kang, Shengtong Zhang, Sang-hyun Kim, Kevin Barreto, Carl Schildkraut, Junehyuk Jung, Jaehyeon Seo, Carlo Pagano, Yuri Chervonyi, Dawsen Hwang, Kaiying Hou, Sergei Gukov, Cheng-Chiang Tsai, Hyunwoo Choi, Youngbeom Jin, Wei-Yuan Li, Hao-An Wu, Ruey-An Shiu, Yu-Sheng Shih, Quoc V. Le, Thang Luong

Aletheia constructed two closed, infinite, countable compact subsets of the complex plane with the same zero transfinite diameter but different values of the polynomial sublevel-area quantity μ(F).

AI involvementAI-autonomous
ImportanceModest
Technical verificationFormal (human)
Human understandingFull
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Sources

  1. Erdős, Herzog and Piranian, Metric properties of polynomials
  2. Feng et al., Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems
  3. Feng and coauthors’ Aletheia paper
  4. Aletheia evaluation report
  5. Later vanishing-question paper
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