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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 1.8 is on p. 3 and its proof in Section 3.2, pp. 7--8. The journal version's pagination and labels were not compared.

Statement

Proposition 1.8 (p. 3). "For any $\varepsilon>0$, there exists $n_0(\varepsilon)\in\mathbb N$ such that if $n\geq n_0(\varepsilon)$, then out of the $m=\Theta(n/\sqrt{\log n})$ distances presented in the $\sqrt n\times\sqrt n$ grid: (i) at least $(1-\varepsilon)\,m/9$ distances occur at least $16n/9$ times; (ii) at least $(1-\varepsilon)\,m/16$ distances occur at least $9n/4$ times; (iii) at least $(1-\varepsilon)\,m/25$ distances occur at least $64n/25$ times."

The paper frames it (p. 3) as sample combinations, not exhaustive, of constants $c_1>0$ and $c_2>1$ for which an $n$-point set with $m$ distances has $c_1m$ distances occurring at least $c_2n$ times, extending Bhowmick's answer to the Erdős–Pach question with $c_1=1/4$, $c_2=1$.

Proof pointer

Section 3.2 (pp. 7--8, Figure 2). The paper proves (ii), with $n=16k^2$: the grid splits into $16$ subgrids of size $k\times k$, each determining $(1\pm o(1))m/16$ distances, and a distance of a non-axis-parallel segment in a subgrid recurs, by translation, at least $36k^2=9n/4$ times in the whole grid; axis-parallel distances are at most $k=o(m)$ in number. It states that (i) and (iii) follow in the same way from $9$ and $25$ subgrids.

Dependencies and read depth

External: the count $(1\pm o(1))cn/\sqrt{\log n}$ of distances in the grid, from Erdős (1946) or Pach and Agarwal, Chap. 12, as cited on p. 7. Read depth: claims checked; the statement and its framing were read clause by clause on the page image of p. 3, and the proof on pp. 7--8 for structure only.

Bears on. #756 (context: multiplicity at least $c_2n$ with $c_2>1$ for a fraction of the grid's $m=\Theta(n/\sqrt{\log n})$ distances, which is $o(n)$ distances, not the $\gg n$ the problem asks for).