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Claim. For every odd $d\ge5$, with $p=\lfloor d/2\rfloor$, there are constants $c_1(d),c_2(d)>0$ such that

$$\frac{p-1}{2p}n^2+c_1n^{4/3}\le f_d(n)\le\frac{p-1}{2p}n^2+c_2n^{4/3}$$

for all large $n$, in the notation of Problem 1085. The leading term is Lenz's, and the second-order term has exact order $n^{4/3}$.

Covers. The estimate of $f_d(n)$ for every odd $d\ge5$: the leading term exactly and the second-order term up to constant factors. The constants $c_1(d)$, $c_2(d)$ and the exact value of $f_d(n)$ are not determined; Swanepoel's structure theorem, on its own claim page in this folder, reduces the exact value for large $n$ to the maximum number of unit distances among $n$ points on a two-sphere, which is open. Nothing is claimed for even $d$ or for $d\le3$.

Depends on. No page of this wiki.

The argument. As Swanepoel's introduction [Sw09] summarizes it, the lower bound improves Lenz's construction in odd dimension by replacing one circle by a two-sphere of radius $1/\sqrt2$ in a three-dimensional subspace orthogonal to the other planes, with points placed on the sphere so that the unit distance occurs at least $cn^{4/3}$ times, a construction of Erdős, Hickerson and Pach; the upper bound combines a stability form of the Erdős–Stone theorem with the $O(n^{4/3})$ bound for unit distances among $n$ points on a two-sphere.

Acceptance. Refereed: P. Erdős and J. Pach, Variations on the theme of repeated distances, Combinatorica 10 (1990), no. 3, 261–269. Not reviewed: the site's remarks credit this result to the paper, but the site labels the problem OPEN, so the remark is not an acceptance of the problem or of a part. The statement is taken from the site's remarks and from the introduction of [Sw09]; the paper has no library card.