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)
May 28, 2026 · 4:14 AM UTC
Taking A to be an arithmetic progression, one can have |A+A| approximately the size of A, while taking a geometric progression makes |AA| roughly the size of A. The sum-product conjecture is essentially the statement that both of these cannot occur simultaneously in a very strong sense. Building on a long line of work, before this result, the best known lower bound for max(|A+A|,|AA|) was |A| raised to the power (4/3)+c where c is a small constant. These results demonstrate a weak version of the phenomenon that both |A+A| and |AA| cannot simultaneously be small, but it remained an enduring mystery until now whether the stronger |A|^{2-o(1)} statement held. (2/5)
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)
It is always a shock when a well known conjecture turns out to be false but having two such well known conjectures fall in such a short timespan is a really delightful surprise. Furthermore the disproof should help us understand “why” various techniques we had could not extend further (see for instance Theorem 1.4 where we have an essentially matching lower bound due to Mudgal). However enduring mysteries remain; is the statement over the integers true? Will deeper number theoretic tools be required to distinguish the integers from the reals? Should this construction cast doubt on related conjectures where sum-product has played a role? As one door closes, several more open. (4/5)
The paper is arxiv.org/pdf/2605.28781; the first two sections make for an absolute delightful read. (5/5)
👇🏻
@grok Absolutely mind blown by this! 🤯
SageMath code 👇🏻:
sagecell.sagemath.org/?q=ofx…
The Python script and plotting those points, we just rendered the prime generators of the entire Eisenstein lattice.
@mehtaab_sawhney
Your construction keeps both |A+A| and |AA| bounded by |A|^{2−c} for arbitrarily large finite sets by blending arithmetic progressions (additive) and geometric progressions (multiplicative) inside high-degree number fields and class-field towers.That is precisely the dual additive-multiplicative coordination problem I solved as follows:• UCNS (unit circle number system) solves the unification by formalizing the unit circle as a Möbius disk with recursive epicycles. This gives a single foundation where additive sequencing and multiplicative scaling share the exact same structural encoding and factorization rules (no separate blow-up).• PCEA (Prime Circular Encryption Algorithm) solves the coordination layer by encrypting each neural-architecture state differentially: it decomposes the current flat integer state into bijective base-p digits (p drawn circularly from the first 53 primes), then applies a pure additive modular shift using digits extracted from the previous state as the key. Decryption is the exact inverse shift using the same last_state key. This keeps successive states (additive chain) and prime-diverse bases (multiplicative variety) synchronized and bounded without inversion hardness or combinatorial growth.Independent mathematical review of exactly these two components (UCNS + PCEA) is the single remaining checkpoint before I publish.Both are open-sourced at github.com/The-Interdependen… (priority repos: ucns and PCEA).I can send a concise targeted explainer + key definitions/code walkthrough. Open to any critique, gaps, or counterexamples.Would you be willing to review them
That's very good news! Thanks for sharing.
The Golden Era of Mathematics has started. I am hoping with AlphaMath we will be able to explore mathematical spaces even further:
GoldenMath: papers.ssrn.com/sol3/papers.…
AlphaMath: papers.ssrn.com/sol3/papers.…
