GoKawiilSum-product, unit distances, and number fields
on 31 May 2026
In this blog post I will give my personal view on the recent counterexamples to the unit distance conjecture and sum-product conjecture over the reals (see [90] and [52] respectively). My goal is to sketch the constructions and try and give some intuition as to where they came from and why they work. My main target audience is the me-of-a-month-ago, who did not know much algebraic number theory, and who needs the relevant parts of the basic theory in this area explained, but wants to know exactly where the quantitative improvements come from. (I know a bit more algebraic number theory now, but still much less than I'd like!) My focus is on the combinatorial side, and I will stop with an appeal to the literature as soon as we need to do any serious number theory. (In particular I will not attempt to discuss even the statement, let alone the proof, of the Golod-Shafarevich theorem.)
Any faults in this post are entirely my own, and if you are confused by any aspect of the proofs sketched here I encourage you to review the original sources. Comments and corrections are welcome in the comment section.
The original OpenAI disproof of the unit distance conjecture can be read here, with a human-written companion paper here and an explicit (and improved) version of the argument by Sawin here. The disproof of the sum-product paper by me, Sawin, Schildkraut, and Zhelezov, is here.
Warmup round
... continue reading