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Tamás Darvas, Binghui Peng and Runzhou Tao, A solution to a problem of Erdős–Herzog–Piranian, a manuscript posted on the Open Science Framework and filed on the site's proof-claims tab on 15 July 2026, and posted in the repository pipeline-math from 15 July 2026 (the move was announced in a comment on the claim on 19 July, and the pinned revision is of 20 July), claims the infimum. Writing $L_f=|\{x\in\mathbb R:|f(x)|<1\}|$ and, for a Borel probability measure $\mu$ on $[-1,1]$ with logarithmic potential $U_\mu(x)=\int\log|x-t|^{-1}d\mu(t)$, $L_\mu=|\{x:U_\mu(x)>0\}|$, Theorem 1.1 states that $\inf_fL_f=\inf_\mu L_\mu=D$, where $D=1.834430475762661711090753635125\ldots$ is defined in the paper's Appendix A as the solution of a system of nonlinear equations. The paper follows Terence Tao's reformulation in terms of potentials: a hypothetical configuration with $L_f<D$ is put in a normal form with one distinguished positive component containing the atom at $-1$, a short interval is chosen to its right, and a dual measure $\lambda$ is built, supported at a point left of that component, at its right endpoint and on the part of the interval where $U_{\mu_f}$ is not positive, with nonnegative potential on $[-1,1]$ and positive mass at a point where $U_{\mu_f}<0$; the symmetry $\int U_\mu d\lambda=\int U_\lambda d\mu$ then gives a contradiction. Section 7 proves sharpness. The supremum $2\sqrt2$ is cited to Elbert's two papers of 1966 and 1968 and to Tao's elementary proof, and the related-work paragraph traces the thread's lower bounds from $1.519$ in December 2025 to $1.814$ in June 2026. Three interval-arithmetic certificates and two exact symbolic verifiers accompany the manuscript. No independent check of the proofs is recorded.

Submission note. Posted to erdosproblems.com as a proof claim by Tam'as Darvas, Bingui Peng, Runzhou Tao (account tdarv) on 15 July 2026, giving "ChatGPT 5.5" as the AI used:

I would like to notify the community of our AI-assisted solution to this problem. Our overall strategy is to finish the argument proposed by Tao in his notes for this problem. We also used contrutions made by users in the comments section. An initial draft of the core argument was generated with the assistance of GPT-5.5 Pro, using the prover–verifier framework (for details see manuscript). The first complete AI-assisted proof was obtained on July 11. Since then, we have substantially revised the argument by simplifying and polishing the proof and we are in the process of identifying appropriate attributions for arguments initially generated by the AI. The entire package is available here: https://osf.io/d6uq5/files/osfstorage?view_only=e31128706d264b9ab0f08692d6702cec Notes: This is my first contribution involving an AI-assisted solution to an Erdős problem. I checked the math, but I had originally intended to spend at least a few more days polishing the manuscript and making a reasonable effort to identify the sources of ideas initially generated by the AI. However, after seeing another update posted today, I am beginning to wonder if this level of caution is no longer the customary practice. We are trying to proceed in a way that is respectful to everyone working on the problem, but it is possible that my expectations are somewhat old-fashioned. If possible, could someone clarify the community guidelines or generally accepted norms for announcing, proposing, or claiming a proof on this forum?

Depends on. [[problems/analysis/E1038/claims/1966_01_01_elbert|Elbert's page]] for the supremum, which the paper cites rather than proves, with Tao's note as the elementary proof and the source of its reformulation. The infimum rests on the paper's own argument.

Standing. The tab lists the authors as Tamás Darvas, Bingui Peng and Runzhou Tao (the paper prints Binghui Peng) and names ChatGPT 5.5; the paper's own disclosure says that an initial version of the main argument was generated by GPT-5.5 Pro in a prover-verifier framework, that the first complete AI-generated proof was obtained on 11 July 2026, and that the authors then verified, revised and simplified it and identified attributions. The submitter's notes ask the community about norms for announcing AI-assisted proofs, and the claim's one comment records the move to the repository. The site's label is unchanged (OPEN), no reviewer is named and nothing is refereed, so the claim stays claimed. The two other claims of the same infimum are Wang's page and Budala's page; the paper notes that Wang's approach is entirely different from its own.