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

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

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

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

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