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

NBNoam Brown@polynoamial

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

NBNoam Brown@polynoamial

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

MSMehtaab 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)

MSMehtaab Sawhney@mehtaab_sawhney

As stated, these results concern the elementary operations of addition and multiplication for real numbers, but closely related statements (“discretized” sum-product estimates) have played a pivotal role in various problems including recent work around the Kakeya conjecture. All previous results seemed to suggest that the original conjecture would hold, and so the new disproof indicates the presence of far richer structure for us to explore. While the original statement of the conjecture was for the integers, essentially all previous results have been extended to the reals and at least to me I never suspected a difference between the two statements. Remarkably the authors demonstrate that the conjecture is false via using constructions coming from increasing degree number fields; the same key ingredient used in the recent disproof of the unit distance conjecture by the internal OpenAI model! In particular the use of class field towers is a crucial ingredient here as well. While I hoped the ideas in the earlier unit distance disproof would yield further fruit, this is beyond my wildest imagination. (3/5)

AKAlex 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.