Sothanaphan 2025 improved lower bound erdos problem concerning

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Nat Sothanaphan, An improved lower bound to Erdos' problem concerning products of distances for fixed diameter. arXiv:2512.14251 (2025). The arXiv record (https://arxiv.org/abs/2512.14251, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

For $n$ points of diameter at most $2$, write $\Delta=\prod_{i\ne j}|z_i-z_j|$ and let $\Delta_{\max}(n)$ be its maximum. The regular $n$-gon has $\Delta=n^n$ when $n$ is even. Proposition 1 (pp. 1--2) proves, with the limit inferior restricted to even $n$, that

$$\liminf_{\substack{n\to\infty\\2\mid n}} \frac{\Delta_{\max}(n)}{n^n} \ge C=\exp\left(-\frac{I\pi^2}{128}\right) \approx 1.0378>1,$$

where

$$I=\int_0^{2\pi}\!\int_0^{2\pi} \frac{\left((1-x/\pi)e^{ix/2}-(1-y/\pi)e^{iy/2}\right)^2 (e^{ix}+e^{iy})}{(e^{ix}-e^{iy})^2}\,dx\,dy \approx-0.481436.$$

In fact the paper constructs, for every even $n$, a configuration with $\Delta=Cn^n(1+O(1/n))$. The perturbation is specified in Section 3.1 (pp. 2--3): antipodal diameter pairs of the roots of unity are pushed and pulled by linearly varying radial amounts. Section 4.1 (p. 3) recasts it as a time-$O(1/n)$ flow under the $n$-independent Lipschitz vector field

$$v(z)=\left(1-\frac{2}{\pi}|\arg z|\right)\frac{z}{|z|}.$$

For $\rho_{ij}=(v(z_i)-v(z_j))/(z_i-z_j)$, Lipschitz continuity bounds all $\rho_{ij}$. The symmetry $v(-z)=v(z)$ pairs $\rho_{ij}$ with $-\rho_{ij}$, cancelling the odd Taylor terms in $\log\Delta$. The quadratic term becomes a Riemann sum for $I$ in Section 4.3.1 (p. 4), where the negativity of $I$, on which $C>1$ depends, is taken from a numerical evaluation (Wolfram Alpha) rather than proved; Section 4.3.2 (pp. 4--5) controls the fourth-order remainder, and Sections 4.3.3--4.3.4 (p. 5) establish $t_{\max}=\pi^2(1+O(1/n))/(4n)$ and combine the estimates to produce $C$.

The result does not treat odd $n$: its construction and antipodal cancellation require even $n$, and Section 2 (p. 2) reports no improvement over the regular odd $n$-gon. Nor does it determine the optimum constant; the paper records a six-arc construction of Cambie, Dong and Tang for $n=6k$, for which numerics suggest $\Delta/n^n$ tends to about $1.30$ but no bound $Cn^n$ with $C>1$ has been proved (Section 2, p. 2), and leaves open whether $\Delta_{\max}(n)/n^n$ tends to infinity along even $n$.

There is a historical caveat to the paper's account of the counterexamples. Danzer and Pommerenke had already disproved regular-polygon optimality for even $n$ in Über die Diskriminante von Mengen gegebenen Durchmessers, Monatshefte für Mathematik 71 (1967), 100--113. Thus this paper's contribution is the explicit asymptotic factor $C>1$, not the first even-$n$ counterexamples.

Source: https://arxiv.org/abs/2512.14251.

The held PDF is the arXiv v1 manuscript (watermark "arXiv:2512.14251v1 [math.MG] 16 Dec 2025" on p. 1), 5 pages, fetched from https://arxiv.org/pdf/2512.14251v1 on 2026-09-23; 316,544 bytes.

Read status. Claims checked: Proposition 1 on the page images (pp. 1--2), the construction and proof outline against the reading copy (pp. 2--5); the proof has not been independently verified.

Bears on. #1045

Results to transcribe.

• Proposition 1 (pp. 1--2): along even $n$, $\liminf\Delta_{\max}(n)/n^n\ge C=\exp(-I\pi^2/128)\approx1.0378$, and an explicit configuration for each even $n$ has $\Delta=Cn^n(1+O(1/n))$.
• Perturbation and integral (Sections 3.1, 4.1, and 4.3.1--4.3.4, pp. 2--5): the linear radial push--pull profile induces the displayed vector field; antipodal symmetry cancels odd variations, the second variation tends to the displayed integral $I$, and the diameter constraint fixes the flow time.
• Scope: the method is even-$n$ only and gives no odd-$n$ improvement; the best constant and divergence of $\Delta_{\max}(n)/n^n$ along even $n$ remain open in this source.
• Historical qualification: Danzer--Pommerenke (1967) already supplied even-$n$ counterexamples, so Proposition 1 is a quantitative asymptotic strengthening rather than the original disproof.