Mathematicians disprove the sum-product conjecture for real numbers using insights from OpenAI's unit distance conjecture counterexample

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After AlphaGo, the skill of human Go players noticeably improved. I suspect we will see a similar pattern in math.

Another major problem, this time in additive combinatorics, has fallen, this time to humans rather than AI, but using methods related to the AI solution to the unit distance conjecture.

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Alex Kontorovich@AlexKontorovich

Now *this* is what I'm talking about! AI giving new ideas, new directions for us humans to pursue. We're entering the "golden era" that Gowers predicted in 1999. (He also predicted it would be short-lived... I hope he's wrong, but I don't know why that would be...)

"The sum-product conjecture is false for real numbers"

By Thomas F Bloom, Will Sawin, Carl Schildkraut, Dmitrii Zhelezov

https://arxiv.org/abs/2605.28781

From the paper:

The role of AI in this proof. The authors were inspired to revisit the possibility of disproving the sum-product conjecture using number fields of large degree by the recent OpenAI counterexample to the unit distance conjecture. Curiously, the final construction given here required far less number theoretic input than the unit distance counterexample.

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Boaz Barak@boazbaraktcs

Wow! I've worked on sum products more than 20 years ago, including using the (true) weaker versions for building randomness extractors.

Love that AI here is not used as a human replacement in "spray and pray" mode for a large collection of open problems, but as a true collaborator.

Mehtaab Sawhney@mehtaab_sawhney

A remarkable paper appeared on arXiv tonight by Thomas Bloom, Will Sawin, Carl Schildkraut and Dmitrii Zhelezov. In this paper, they prove that there exists c>0 and arbitrarily large finite sets A of real numbers such that max(|A+A|,|AA|)≤|A|^{2-c}. This disproves the well-known sum-product conjecture over the real numbers. The sum-product conjecture considers the two most basic operations: addition and multiplication. A+A is the set of all pairwise sums of two elements in A while AA is the set of all pairwise products of two elements in A. (1/5)

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anpaure@anpaure

@polynoamial not true according to a go player, skill rose because people just memorized lines further, starting from move 60 the move accuracy is similar to the past https://www.lesswrong.com/posts/nR3DkyivzF4ve97oM/how-go-players-disempower-themselves-to-ai

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Noam Brown@polynoamial

Source: https://www.henrikkarlsson.xyz/p/go

Noam Brown@polynoamial

After AlphaGo, the skill of human Go players noticeably improved. I suspect we will see a similar pattern in math.

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David Bessis@davidbessis

@polynoamial The rise of intuition-maxxers. https://davidbessis.substack.com/p/the-fall-of-the-theorem-economy

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Tom McMurtry@TomMcMurtryNZ

@wtgowers Didn't the chess people also use chess engines to learn from and improve their play for a little bit beyond usual and beyond the engines? Then the chess engines got too much better

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Boris Power@BorisMPower

@anpaure @polynoamial I’m a Go player too and know many professional players. The quality of play improved significantly, as professionals can now review their games and get very strong feedback.

A similar thing happened in chess.

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anpaure@anpaure

useful to know! the author himself is 7d, so barely below professional ranking, smth like 2400-2500 fide in chess terms, definitely not elite level but i assume he'd be knowledgable about top level play and prep? either way, it's helpful to read the whole post, and this other post specifically talking about improvement https://tuesdaybornwhale.github.io/posts/learning_from_superhumanai/

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Martin@Martin778192374

@wtgowers It is funny to think about how much flack AI got for solving new problems using old results "oh it just re-used and combined X, Y, Z and brute-forced => easy". Imagine if this standard was applied to this paper and people went "Not really new, they just used AI ideas" ;-)

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Robin East@Insidedctm

@wtgowers Time to introduce the Lump of Math Fallacy. Inspired by Lump of Labour Fallacy, it’s the (mistaken) idea that there is some fixed amount of maths to be solved and eventually machines will just solve it, putting human mathematicians out of work.

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Peter Gostev@petergostev

@polynoamial This data cuts off at 2021, I tried to replicate this / extend the dataset, couldn't do it exactly, but looks like the effect has about evened out at this higher level, ie there was no further growth beyond this. Which is kind of interesting too

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Alex Kontorovich@AlexKontorovich

@davidmanheim I don't know where the twitter thread is anymore, but I discuss it here (towards the end): https://arxiv.org/abs/2510.15924

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Aryaman@MishraArya5819

@TomMcMurtryNZ @wtgowers Even today best chess players learn from chess engines to become stronger.

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TKD@tkdprotocol

@wtgowers kinda wild that the big breakthroughs lately are coming from people borrowing the ai playbook without the ai

makes me wonder how many “impossible” problems were just waiting for a different lens the whole time

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Tim Isgro@TimIsgro

@anpaure @polynoamial Isn't this a little too pessimistic? Couldn't you make the same argument with chess players (that players today memorize openings more so than in the past)? But I don't think people argue that the improvement in chess ELO is fake.

Genuinely curious.

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Joe Shipman@joe_shipman

@TomMcMurtryNZ @wtgowers Yes but to a MUCH smaller degree than in Go. Human grandmasters had developed chess theory very well over the previous 200 years, they understood the game a lot better than Go players understood their game. Also they actually learned more from the most recent AI-fueled engines.

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Psyho@FakePsyho

@polynoamial *as long as humans remain relevant in math

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claude shannon@catpoopburglar

@polynoamial almost all of this improvement is either cheating or memorization of AI move patterns

https://tuesdaybornwhale.github.io/posts/learning_from_superhumanai/?utm_source=chatgpt.com

it’s notable that in the best officiated tournaments, there has been no improvement after move 60

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David Manheim@davidmanheim

@polynoamial Agreed - but my question is about how the centaur model progresses going forward. For Chess, there were several years, but for Go, that centaur-dominant period, it if existed at all, was very brief. @wtgowers

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