The Unit Distance Problem

In which I delve into the actual math

There’s a new math result which is a milestone for AI mathematics. It’s a human readable and insightful result on a conjecture of some renown. It improves on a previous construction of Erdos to make a set of points in the plane with a relatively large number of unit distances between them.

Where the AI got its inspiration from can be as ineffable as it is for humans, but there’s a plausible narrative that it got direct inspiration from the Erdos construction. A proof tells a story, and the moral of the story belongs to the reader not the storyteller. To some the Erdos construction is a story about square grids. But it can also be read as a story about taking an algebraic construction, finding a projection onto geometric space which preserves unit distances, and then solving a number theory problem in the algebraic space to have lots of unit distances. Instead of using the straightforward grid structure the new construction uses a more esoteric algebraic construction, involving pulling in a powerful theorem from a completely different place. In a funny detail the underlying number theory problem it relies on is fairly trivial while the Erdos one requires some work. That is not coincidental with there being a lot more edges: the requirements for them to work are much less stringent.

The obvious question is: What does it look like? The papers and articles contain no pictures of the new construction and there’s a reason for that but another reason one should be included anyway. The construction used for small examples produces some very tesseract-looking things and at larger scales looks like a point cloud without any obvious nice geometric properties. At the smaller scale where the structure can be gleaned it looks actively counterproductive, producing fewer distance coincidences than the Erdos construction. You have to crank up the number of dimensions and the radius of the ball up quite a bit before it starts getting favored, and by then the number of points has become huge.

But that doesn’t mean there can’t be a picture! You can have a density plot where regions with more points points are darker, and having the picture may yield geometric insights which the algebraic construction was obfuscating. Does it look like the shadow of a sphere? A disc? A Gaussian plot? Whatever the shape is, the next question is: How big is the unit distance compared to the width of the shape? Here is where it gets interesting: It appears to be that the distance is quite small. For me that starts raising alarm bells. Didn’t we already crop to within a ball in the algebraic construction? Yes we did, but that was to make the number of points finite, not to reduce the geometric range. The projection between the algebraic and geometric space makes many things look very different with the one exception that certain exactly unit distances stay unit. Other distances get scrambled. So that raises the next question: Why can’t we just crop geometrically to some small constant factor of the unit distance at the end, thus making a much better result by reducing the denominator? This might actually work! It depends on just how much smaller the cropping is and how sparse of a region can be found. I honestly don’t know if it works out, and don’t have the tools to analyze this because it’s a bizarre jump back into geometric space from algebraic but it’s plausible and the benefits might be big, so it’s certainly worthy of further analysis.

The concrete bounds now stand at there being a lower bound on the polynomial exponent of 1.014, up from the previously conjectured to be optimal value of 1. The known upper bound is 4/3. That range of possibilities is very interesting and we most definitely have not heard the last word on this. The AI construction just showed 1+e and the 1.014 is a later explicit improvement. Maybe there will be a polymath project on it.

Talking to AI (specifically Opus 4.7) about this is very interesting. It can read through the whole construction no problem, and talk about it fluently. But then when it gets into discussing geometric insights its intuition is garbage. With some prodding I can get it to understand basic points, and it readily understand after they’re pointed out that these are very basic things, but it just can’t wrap its brain around anything without having it explained. It seems like the new construction is exactly the thing it happens to be super good at: Tackling something purely symbolically, pulling in outside theorems and constructions from seemingly totally unrelated areas, following a roadmap which had already been laid out for it. Drawing from geometric intuitions is something which it simply can’t do. The contrast is very bizarre in this particular case where it’s going from genius to idiot talking about the exact same problem with the perspective shifted only slightly. I haven’t, and probably won’t, grok the full new construction, but it was able to explain the basics outline of the construction to me and construct some basic examples, which was fun and interesting.

The other notable thing about the AI strength here is that this is a constructive proof. AI seems to be better at that than proofs of nonexistence, which is consistent with it being fast and not having much insight. Constructions require fiddling around until you find something, with much clearer partial results along the way, where with proofs of non-existence you have to intuit a roadmap or you don’t make any obvious headway until the very end. The proof of the Robbins conjecture is similar: The core insight is up front realizing that you can find a counterexample to Modus Tollens and then do proof by contradiction. After that it looks a lot more like finding a solution to a post substitution problem than a meaningful proof.