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Claim. Let $M(n)$ be the maximum, over all $n$-point sets $A\subset\mathbb R^2$, of the gap $f(d_1)-f(d_2)$ between the two largest distance multiplicities of $A$, the quantity Problem 959 asks to estimate. Felix Christian Clemen, Adrian Dumitrescu and Dingyuan Liu, On multiplicities of interpoint distances, Acta Math. Hungar. 177 (2025), no. 1, 231-245, cited as [CDL25] on the problem page (library home clemen_2025_multiplicities_interpoint_distances), prove $M(n)=\Omega(n\log n)$ (Corollary 1.10). The corollary is the case $k=1$ of their Theorem 1.9: for every sufficiently large $n$ and every $1\le k\le\log n$ there is an $n$-point planar set with $f(d_k)-f(d_{k+1})=\Omega\bigl((n/k)\log n\bigr)$, and the distances with the $k$ largest multiplicities can be prescribed. Their Problem 1.11 asks whether $M(n)\ge n^{1+c/\log\log n}$ for some $c>0$ and all large $n$.

Covers. A lower bound of order $n\log n$ on $M(n)$, and the gaps $f(d_k)-f(d_{k+1})$ for $k\le\log n$. No upper bound on $M(n)$ is proved, and the order of $M(n)$ remains open.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper is the publisher's version of record in Acta Mathematica Hungarica, volume 177 (2025), pages 231-245, published online on 12 November 2025; the preprint arXiv:2505.04283 was first posted on 7 May 2025, the date this page carries. Not reviewed under the corpus's rule: the site's commentary credits [CDL25] with the $n\log n$ bound, but the site labels the problem OPEN, so that commentary is not an acceptance that settles it. The later claims of larger gaps are Snyder's page and Xeff's page.