---

Source. M. Charalambides, A note on distinct distance subsets, J. Geom. 104 (2013), no. 3, 439--442, DOI 10.1007/s00022-013-0176-0; read in the arXiv preprint arXiv:1211.1776v1, whose labels and page numbers are used here. Proposition 1.2 and the sentence deriving it on p. 1. The journal version was not compared. The edition read is identified on the source card.

Statement

With $\delta(N)$ the minimum, over $N$-point sets $P\subset\mathbb R^2$, of the largest subset of $P$ with all pairwise distances distinct (see Proposition 2.1), and $X\lesssim Y$ read as $X\le CY$ for an absolute constant $C$ (the paper uses the notation without defining it):

Proposition 1.2 (p. 1). "$\delta(N)\lesssim N^{1/2}(\log N)^{-1/4}$"

The paper derives it in its introduction from a one-sentence grid argument; the abstract announces only the lower bound. The paper attributes the question of the order of $\delta(N)$ (Question 1.1, p. 1) to Avis, Erdős and Pach.

Proof pointer

The paper's justification is one sentence (p. 1): a $\sqrt N\times\sqrt N$ integer grid determines $\lesssim N/\sqrt{\log N}$ distinct distances. The paper leaves the deduction implicit: a subset of size $m$ with all distances distinct uses $\binom m2$ different distances, so $\binom m2\lesssim N/\sqrt{\log N}$ and $m\lesssim N^{1/2}(\log N)^{-1/4}$. The paper states the grid's distance count without proof or reference.

Coverage

Claims checked: the statement and its one-sentence derivation were read on the page image of p. 1. The grid's distance count was not checked. Nothing here is independently reviewed.

Bears on. #1208, for $d=2$: $\delta(N)$ is that problem's $F_2(N)$, so this is the upper bound $F_2(N)\lesssim N^{1/2}(\log N)^{-1/4}$. With the lower bound of Proposition 2.1 it leaves the order of $F_2(N)$ undetermined.