Erdős distinct-distances problem in R3

The problem asks whether every set of $N$ points in three-dimensional space determines at least about $N^{2/3}$ different distances, matching the integer-lattice construction. The conjectured exponent remains the benchmark for the unrestricted problem.

Known results

• The integer lattice gives an upper bound of order $N^{2/3}$ for the minimum possible number of distances.
• Aronov, Pach, Sharir, and Tardos (2004) obtained a lower bound of roughly $N^{0.546}$.
• Solymosi and Vu improved the unrestricted three-dimensional lower bound to $N^{3/5}$ up to logarithmic factors.
• Stronger bounds are known for points constrained to suitable algebraic surfaces, but not for arbitrary point sets.

August 2026 near-optimal bound

Jonathan Tidor, Hung-Hsun Hans Yu, and Dmitrii Zakharov claim that every $N$-point set determines at least $N^{2/3-o(1)}$ distances, quantitatively with an error of order $\sqrt{\log\log N/\log N}$. This reaches the conjectured exponent up to a subpolynomial factor, but the preprint does not prove the exact $N^{2/3}$ bound and has no independent verification in the supplied sources.

Current status (as of August 2026): The best reported general bound is the unverified preprint claim $N^{2/3-o(1)}$, while the exact conjectured order $N^{2/3}$ remains open.