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Source. F. C. Clemen, A. Dumitrescu and D. Liu, On multiplicities of interpoint distances, Acta Math. Hungar. 177 (2025), no. 1, 231--245, DOI 10.1007/s10474-025-01562-y; read as arXiv:2505.04283v5 (3 February 2026), whose printed page numbers equal its PDF pages. Proposition 1.5 is on p. 2, Problem 1.6 on p. 3 and the proof in Section 2.3, p. 6. The journal version's pagination and labels were not compared.

Statement

Notation as on p. 2: $\Delta_2=\Delta_2(X)$ is the second largest and $\delta=\delta(X)$ the smallest distance in $X$, and $\mu(X,d)$ the multiplicity of $d$.

Proposition 1.5 (p. 2). "Let $m,n\in\mathbb N$ with $m\leq\lfloor n/2\rfloor$. There exists a planar point set $X$ with $|X|=n$, such that $\mu(X,\Delta_2)\geq3m$ and $\mu(X,\delta)\geq3n-5m+o(m)$."

The count at the end of the proof (p. 6) is written $3n-5m+o(n)$. Taking $m=\lfloor3n/8\rfloor$, the paper obtains (p. 2) $\min\{\mu(X,\Delta_2),\mu(X,\delta)\}\geq9n/8+o(n)$.

Problem 1.6 (p. 3). Determine

$$\limsup_{n\to\infty}\ \sup_{X\subseteq\mathbb R^2,\,|X|=n}\ \frac{\min\{\mu(X,\Delta_2),\mu(X,\delta)\}}{n}.$$

The paper's motivation (p. 2): one way to settle Conjecture 1.1 would be to show that one of two chosen distances always occurs at most $n$ times, and the proposition shows that the smallest and the second largest distance cannot serve as that pair.

Proof pointer

Section 2.3 (p. 6, Figure 1), after Vesztergombi's construction. With $m_1=m_2=m$ and $m_3=n-2m$: a regular $m$-gon inscribed in a circle of radius $n$; $m$ points inside it, on a circle, each joined to two polygon vertices at the polygon's second largest distance, consecutive ones at a distance $\delta=\Theta(1)$; and $n-2m$ points of a triangular lattice of mesh $\delta$ in a disk of radius $\Theta(\sqrt n)$ at the center. The first two groups are the two convex layers.

Dependencies and read depth

External: Vesztergombi (1987, 1996) and Braß, Moser and Pach, Chap. 5.8, for the construction's model, as cited on p. 6. Read depth: claims checked; Proposition 1.5, the $9n/8$ deduction and Problem 1.6 were read clause by clause on the page images of pp. 2--3, and the construction on p. 6 for structure only.

Bears on. #132: a limitation on one route to the first question (the smallest and the second largest distance can both occur more than $n$ times); it settles no case of the problem.