Pack points onto a plane and a new construction gets more pairs exactly one unit apart than Erdős thought possible.
How close is the n^1.014 bound to the known upper bound of n^4/3?
This asks how far the new n^1.014 lower bound for unit distances (the most pairs of points exactly one unit apart that anyone has built) is from the n^4/3 upper bound (the most that can possibly exist), and what the remaining gap means.
This asks how far the new n^1.014 construction is from the n^4/3 ceiling, and whether the gap is closing. The short answer is that it is still very far. Measured by exponents, the problem sits between 1 (roughly one unit-distance pair per point) and 4/3 (the long-standing upper bound from Spencer, Szemerédi and Trotter, which comes from general mathematics knowledge rather than this collection). The new construction moves the lower end from 'essentially 1' to 1.014. That covers about 4% of the distance from 1 to 4/3. For a set of a million points, n^1.014 is about 1.2 million unit-distance pairs, while n^4/3 is about 100 million. So the construction barely moves the gap. Its importance is somewhere else.
The importance is that it disproves Erdős's conjecture. Erdős expected the true answer to be only slightly more than linear, growing more slowly than n raised to any fixed power above 1. Any exponent above 1, even 1.014, shows that guess was wrong. The collection traces how this was done. OpenAI's counterexample built planar point sets from classical number-theory tools (Golod–Shafarevich towers and number fields). The new ingredient was letting the degree of the number field grow without limit, which keeps certain quantities that would otherwise grow out of control under control (What made OpenAI's unit distance counterexample succeed?). That first result proved some exponent above 1 exists but didn't say what it was. A follow-up lattice construction made it explicit at about 1.014 (How many unit distances can points in a plane have?). Its summary gives the figure as 1.014114 with an absolute constant attached, so read '1.014' as a proven floor rather than a tight value.
The gap leaves an open question: is the true answer near 1, near 4/3, or somewhere in between? Nothing in the collection settles it. The square-grid constructions that suggested near-linear growth now look too pessimistic. On the other side, nobody has shown that the 4/3 ceiling, which comes from general incidence-counting arguments, is reached in the plane. The upper bound may be loose, the lower bound may be weak, or both.
There is also a lesson about how AI-assisted mathematics moves forward. Tao argues that a machine-learning system being opaque matters less when its output can be checked by something reliable, such as a proof assistant or a rigorous argument (Can opaque machine learning models help prove new mathematics?). The unit-distance story fits that pattern. The AI-linked result showed that a particular construction method works, and conventional mathematics then turned it into an exact number. Expect further progress the same way: better exponents from tuning the number-field construction, not one jump to 4/3.
The collection is thin here. It covers how the counterexample was built and how the exponent was made explicit. It doesn't cover the upper-bound side or whether 4/3 can be improved, so the comparison above draws on standard background for the ceiling.
Sources 3 notes
OpenAI's counterexample to Erdős's unit distance conjecture builds on classical Golod–Shafarevich towers and number-field methods, but achieves its breakthrough by letting the degree [K : Q] → ∞. This allows a fixed split prime to suppress the class number and discriminant, enabling the construction of planar point sets with superlinear unit distances.
A lattice-based construction using CM fields and Golod-Shafarevich arguments produces sets of n points with more than n^1.014 pairs at unit distance, making explicit the exponent that a prior OpenAI result left unspecified. The bound is stated as 1.014114/C for an absolute constant C.
Tao argues ML tools' opacity matters less than pairing them with reliable validators like proof assistants or numerical methods. He cites finite-time blowup for Boussinesq equations, where a neural network suggested solutions later verified through perturbation arguments.
Papers this line draws on 8
The research behind the notes this line reads — ranked by how closely each paper relates.
- Verification abundance, adjudication scarcity: what happens to mathematical knowledge when proof checking becomes free
- From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier
- 'hello there the jacobian conjecture is false thanx': why a tiny social media post has mathematicians rethinking AI
- Remarks on the disproof of the unit distance conjecture
- An explicit lower bound for the unit distance problem
- The crisis of AI-generated mathematics
- Machine-Assisted Proof
- We'll Be Arguing for Years Whether Large Language Models Can Make New Scientific Discoveries