The Disproof of Unit Distance Conjecture

Thu, 28 May 2026 17:13:21 -0400

After learning about the recent OpenAI disproof of the unit distance conjecture ⟦cite:OpenAI26⟧ , I want to devote some time to understand some main ideas in this paper. The problem goes back to Erdős's 1946 paper ⟦cite:Erd46⟧ , and the classical upper bound is due to Spencer, Szemerédi, and Trotter ⟦cite:SST84⟧ . There are new developments by Will Sawin, who gave a more concrete bound ⟦cite:Saw26⟧ , and there is also a human-verified expository note with expert comments ⟦cite:ABG+26⟧ . For broader background, see Brass, Moser, and Pach ⟦cite:BMP05⟧ .

Main Ideas

For a finite set $P\subseteq\mathbb C$, let

$$\nu(P):=\#\{\{x,y\}\subseteq P:|x-y|=1\}\qquad \nu(n)=\mathrm{max}_{\# P=n}\nu(P)$$

Erdos conjectured that $\nu(n)=n^{1+o(1)}$. The result that was shown was the following

(OpenAI) There exists some $\delta>0$ such that $\nu(n)>n^{1+\delta}$ for infinitely many $n$.

This disproves Erdos. Let $f\ge 1$, then in $\mathbb C^f$, denote the polydisc

$$B_R=\{(x_1,\dots,x_f)\in\mathbb C^f:\forall i,|x_i|\le R\}$$

and for a (full) lattice $\Lambda\subseteq\mathbb C^f$, define

$$U_{\Lambda}=\{(x_1,\dots,x_f)\in\Lambda:\forall i,|x_i|=1\}$$

Let $\pi:\mathbb \Lambda\rightarrow\mathbb C$ be the projection to an arbitrary coordinate and and assume it is injective. If $\Lambda$ can be found such that $U_\Lambda$ is large, then for $R>1$, the set $P_{\Lambda,R}:=\pi(U_{\Lambda}\cap B_R)$ is a finite set of points with at least $\frac{1}{2}|U_{\Lambda}||\Lambda\cap B_{R-1}|$ unit distance pairs among at most $|U_{\Lambda}\cap B_{R}|$ points. One of the core lemmas in the paper proves an explicit estimate for this type of statement with a translate $a+\Lambda$ of the lattice, and it turns out this estimate is enough to produce the lower bound.

Take a totally real number field $F$ of degree $f$ and form the CM field $K/F$ by quadratic imaginary extension, one can from the Minkowski embedding $K\rightarrow\mathbb C^f$ and form the lattice $\Lambda$ by taking the image of a fractional ideal. If we have find $K$ such that many primes $\mathfrak q$ in $F$ split, then we can find many $u=\frac{\mathfrak q}{\overline{\mathfrak q}}$ of unit norm, so that $U_{\Lambda}$ is big enough. Such fields $K$ are built from Golod–Shafarevich class field towers, which is a tower of CM fields, where ramification is controlled.

References

[OpenAI26]OpenAI. Planar Point Sets with Many Unit Distances. 2026. https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf.
[ABG+26]Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, and Melanie Matchett Wood. Remarks on the Disproof of the Unit Distance Conjecture. 2026. arXiv:2605.20695. https://arxiv.org/abs/2605.20695.
[Saw26]Will Sawin. An Explicit Lower Bound for the Unit Distance Problem. 2026. arXiv:2605.20579. https://arxiv.org/abs/2605.20579.
[Erd46]Paul Erdős. On Sets of Distances of $n$ Points. American Mathematical Monthly. 53(5). pp. 248--250. 1946.
[SST84]Joel Spencer, Endre Szemerédi, and William T. Trotter. Béla Bollobás (ed.). Unit Distances in the Euclidean Plane. In Graph Theory and Combinatorics. pp. 293--303. Academic Press. 1984.
[BMP05]Peter Brass, William Moser, and János Pach. Research Problems in Discrete Geometry. Springer. 2005.

Discrete-Geometry on Yunhai Xiang

The Disproof of Unit Distance Conjecture

Main Ideas

References