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Claim. Jonathan Tidor, Hung-Hsun Hans Yu and Dmitrii Zakharov, The Erdős distinct distances problem in $\mathbb{R}^3$, arXiv:2608.14454, version 1 of 14 August 2026, prove that every set of $N$ points in $\mathbb{R}^3$ determines at least $N^{2/3-o(1)}$ distinct distances. In the notation of Problem 1083 this is $f_3(n)\ge n^{2/3-o(1)}$; with Erdős's upper bound $f_3(n)\ll n^{2/3}$ from the integer grid [Er46b] it gives $f_3(n)=n^{2/3-o(1)}$, which is the particular question of the problem, answered yes, for $d=3$. The previous lower bounds in three dimensions were $n^{1/2}$ (Clarkson, Edelsbrunner, Guibas, Sharir and Welzl), $n^{0.546}$ (Aronov, Pach, Sharir and Tardos) and $n^{3/5}$ (Solymosi and Vu combined with the planar bound of Guth and Katz; $n^{3/5}/(\log n)^{2/5}$ in the release preprint's statement), as the site's remarks record them. The result is also recorded, as general-space context, on the page of Problem 660.

Covers. The case $d=3$ of the question whether $f_d(n)=n^{2/d-o(1)}$. Nothing is claimed for $d\ge4$, and the $o(1)$ in the exponent is not removed; the release preprint recorded on [[problems/distance_problems/E1083/claims/2026_09_23_openai|OpenAI's claim page]] claims the constant-factor bound $f_d(n)\gg_d n^{2/d}$ for every $d\ge3$, which would supersede this result.

Depends on. No page of this wiki.

Acceptance. None documented. The result is an arXiv preprint with no journal publication recorded; a poster reported it on the site's thread on 17 August 2026 as solving the case $d=3$ of the problem, and the site's page, last edited 16 October 2025 and labeled OPEN, does not mention it, so there is no curator credit. The claim is claimed.