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Statement

Setting (p. 1). For a finite point set $P$, $u(P)$ is the number of unordered pairs of points of $P$ at Euclidean distance $1$. For $D>1$, $\mathbb S^2_D$ is the sphere in $\mathbb R^3$ of diameter $D$ centred at the origin, and for $n\ge1$, $u_D(n)=\max\{u(P):P\subset\mathbb S^2_D,\ \#P=n\}$.

Theorem 1 (p. 1, quoted). "There exists $c>0$ such that for any $D>1$ and $n\ge2$, $u_D(n)>cn\sqrt{\log n}$."

The constant $c$ is one constant for all $D>1$ and all $n\ge2$. The proof (p. 3) remarks that $c=1/10$ serves when the logarithm is to base $2$, and that the method gives the asymptotic form $(1-o(1))\tfrac12 n\sqrt{\log_2 n}$; the displays there write $u(n)$ for the quantity being bounded.

The paper places Theorem 1 against Leo Moser's conjecture that $u_D(n)<cn$ for every $D>1$, and against Erdős, Hickerson and Pach (1989), who disproved it with $u_{\sqrt2}(n)=\Theta(n^{4/3})$ and $u_D(n)>cn\log^*n$ for all $D>1$ and $n\ge2$, $\log^*$ the iterated logarithm (p. 1). It records $u_D(n)<cn^{4/3}$ as the best known upper bound, of the right order for $D=\sqrt2$ and with nothing more known for other $D>1$ (p. 2). It also states, without proof, that Theorem 1 holds for the hyperbolic plane of any curvature with a virtually identical proof, an observation it credits to Endre Makai Jr. (p. 2).

Proof pointer

Section 2 (pp. 2--4). Rotations about the axis through the poles act on $\mathbb S^2_D$ as an abelian group of isometries. Take a set $A$ of $t\ge1$ points in a small neighbourhood of a point of the equator; for an ordered pair $(p,q)\in A^2$ let $\beta(p,q)$ be the counterclockwise angle with $|pq_{\beta}|=1$, $q_\beta$ the image of $q$ under rotation by $\beta$, and for $S\subseteq A^2$ let $\beta(S)$ be the sum of $\beta(p,q)$ over $(p,q)\in S$.

• Claim 1 (p. 2): for every $t\ge1$, $A$ can be chosen so that the $2^{t^2}$ rotated copies $A_{\beta(S)}$, $S$ ranging over the subsets of $A^2$, are pairwise disjoint.
• Given Claim 1, the union $B$ of these copies has $t2^{t^2}$ points, and two index sets differing in one pair give a unit distance between their copies, so $u_D(t2^{t^2})\ge\tfrac{t^2}{2}2^{t^2}$ (p. 2). For general $n$, take the $t$ with $t2^{t^2}\le n<(t+1)2^{(t+1)^2}$, disjoint slightly rotated copies of $B$ and arbitrary extra points (pp. 2--3).
• Claim 1 is proved (pp. 3--4) by small perturbations of $A$ that make every pair of index sets $\{S,S'\}$ satisfy $\beta(S)\ne\beta(S')$, using Observations 1--4 and Claim 2 (p. 4), which moves two points of $A$ along circles so as to change $\beta(a,b)$.

Read depth

Claims checked: the definitions, Theorem 1 and Claim 1 were read clause by clause on the page images of the author version named on the source card, and the proof in Section 2 was followed. Nothing here is independently reviewed.

Dependencies

None in the corpus. The proof in Section 2 cites no other result.

Source. K. J. Swanepoel and P. Valtr, The unit distance problem on spheres, in Towards a Theory of Geometric Graphs, Contemp. Math. 342, Amer. Math. Soc., Providence, RI, 2004, 273--279, doi:10.1090/conm/342/06148; page numbers refer to the author version named on the source card.

Bears on

• Problem 605: the problem asks for $n$ points on a two-dimensional sphere with at least $f(n)n$ pairs at one common distance, for some $f(n)\to\infty$. Theorem 1 gives, on the sphere of diameter $D$ for each $D>1$, $n$ points with more than $cn\sqrt{\log n}$ pairs at distance $1$, so $f(n)=c\sqrt{\log n}$ is a function of that kind; it does not determine the order of growth of the maximum.