---

Terence Tao, Sublevel sets of logarithmic potentials, a note dated 20 December 2025 and posted to the problem's thread on 21 December 2025, proves the supremum half of the problem. For a probability measure $\mu$ on $\mathbb R$ with logarithmic potential $U_\mu(x)=\int\log|x-t|^{-1}\,d\mu(t)$, Theorem 1.1 states that if $\mu$ is supported in $[t_0-1,t_0+1]$ then $|\{x:U_\mu(x)\ge0\}|\le2\sqrt2$, the bound being sharp for $\mu=\frac12\delta_{-1}+\frac12\delta_1$; Corollary 1.1 applies it to the root measure of a nonconstant monic polynomial $P$ with all roots real in $[t_0-1,t_0+1]$ and gives $|\{x:|P(x)|\le1\}|\le2\sqrt2$, which the note says resolves the question of Erdős, Herzog and Piranian. The proof rests on a duality lemma, that a finite measure whose potential is negative on the support interval cannot have its support inside the sublevel set, a monotone measure-preserving rearrangement and three explicit trial measures, the parameter values of the second and third of which the note says were supplied by AlphaEvolve. An expanded version dated 22 December 2025 and posted to the thread the same day first states the result as Theorem 2.1 for probability measures, with the characterization of the equality case, $\mu=\frac12\delta_{t_0-1}+\frac12\delta_{t_0+1}$; the note Superlevel sets of logarithmic potentials, dated 27 December 2025 and posted to the thread on 28 December, restates that theorem and adds structural reductions and a one-cut candidate for the infimum, which it does not determine. No independent check of the proofs is recorded.

Submission note. Posted to the site's forum by Terence Tao on 21 December 2025:

Here is a writeup (changing some sign conventions etc. to streamline the proof a bit) which I think completes the proof that the supremum is $2\sqrt{2}$. To treat the intermediate regime $1.7624 < M < 1.7987$ I got AlphaEvolve (following a suggestion of natso26) to suggest the weight

$$> \delta_0 + A 1_{[a,2\sqrt{2}]}\ dx + B 1_{[b,2\sqrt{2}]}\ dx + C \delta_{2\sqrt{2}} >$$

(shifted by $-M$) with $A = 0.192829$, $B = 0.224$, $C = 0.155$, $a = 1.63$, $b = 1.919$. But this is far from the only choice, there are a lot other ways to proceed here (I have not checked jspier's alternate approach carefully but it looks plausible).

(The site has been updated to address this comment.)

Posted to the site's forum by Terence Tao on 22 December 2025:

Here is an expansion of the previous notes. It achieves step 1 (existence of the minimizer) but not step 2 or step 3. Using some arguments of Erdos, I could reduce to the case where the $\{U_\mu > 0 \}$ contains a large interval $I \supset (-\sqrt{2},0]$, in which $\mu$ consists of a Dirac mass at $-1$ of measure at least $1/2$; but I could not exclude the possibility that $\{U_\mu > 0 \}$ also contains other intervals (though in each such interval, $\mu$ can be assumed to be a Dirac mass).

Covers. The supremum only: $\sup|E_f|=2\sqrt2$ for $E_f=\{x\in\mathbb R:|f(x)|<1\}$ over the class, with the equality case from the version of 22 December onward. The claim says nothing about the infimum beyond the reductions and the candidate value, which the three full claims of the folder build on.

Standing. The note is a dated manuscript on the author's site, not a refereed paper and not filed on the proof-claims tab. Darvas, Peng and Tao describe it as the elementary proof Erdős asked for of the theorem on Elbert's page, and Wang's Theorem 10.1 imports Theorem 2.1 of the updated note for the supremum. The formal-conjectures statement file, at its revision of 2026-09-18 (1038.lean), tags its supremum part research solved and cites the note for it, with a link to the version of 22 December; a statement file is not a formalization link. The site's label is OPEN and its commentary lists the supremum among the known values, thanking Terence Tao among the contributors without crediting a proof, so no reviewer is named and the claim stays claimed.

Depends on. Nothing on the wiki.