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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 5.3 is on p. 9 and its proof on pp. 9--10; Observation 5.4 is on p. 10. The journal version's pagination and labels were not compared.

Statement

Proposition 5.3 (p. 9). "For every $n\in\mathbb N$, there is a set $X\subseteq\mathbb R^2$ of $n$ points with pairwise distinct distance multiplicities and $a(X)\neq(n-1,n-2,\ldots,1)$."

It answers the second question of Problem 5.2 (p. 9), whether the configurations of Figure 3 (equidistant points on a line or a circle, and Observation 5.1's arc with its center) are the only ones with pairwise distinct distance multiplicities, in the negative. The paper also shows that an integer grid is not a candidate: Observation 5.4 (p. 10) states that for $k\ge4$ the $k\times k$ grid has two distances that each occur exactly $8$ times.

Proof pointer

Pages 9--10, Figure 4. For odd $n=2k+1$: $k+1$ and $k$ points of the hexagonal lattice of side length $1$, as the paper calls it, on two adjacent horizontal lines. The distance $1$ occurs $4k-1$ times, the integer distance $j$ occurs $2(k-j)+1$ times for $2\le j\le k$, and the distance $d_j=\sqrt{j^2+j+1}$ occurs $2(k-j)$ times for $1\le j\le k-1$; these multiplicities are pairwise distinct. The paper states that even $n$ is analogous and leaves it to the reader.

Dependencies and read depth

None external. Read depth: claims checked; Proposition 5.3, the multiplicity list in its proof and Observation 5.4 were read clause by clause on the page images of pp. 9--10.

Bears on. #958 (context only: it concerns sets whose multiplicities are merely distinct, not the profile $(n-1,\ldots,1)$ the problem characterizes).