The sum-product conjecture is false for real numbers, disproved via constructions of sets A with max(|A+A|, |AA|) ≤ |A|^{2-c} for absolute c > 0.
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Remarks on the disproof of the unit distance conjecture
13 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
We present a short, digested, human-verified version of the recent OpenAI-generated counterexample to the Erd\H{o}s unit distance conjecture, and a sequence of reflections on it. The argument relies crucially on ideas that may, at least in retrospect, be attributed to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna.
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2026 13representative citing papers
Constructs sets A subset R with |{x+y+(x-y)^2 : x,y in A}| <= |A|^{2-c} for some c>0, giving a counterexample to the Elekes-Rónyai problem via prime-splitting amplification.
Constructs lattice point sets with many rectangles and few isosceles triangles to produce explicit counterexamples to the Mizohata-Takeuchi conjecture for the paraboloid via transference principles.
Proves R(B_8, B_10) = 37 via an AI-assisted short proof with a Lean formalization of the upper bound.
Introduces an auditable four-stage diagnostic for LLM physics reasoning in novel frameworks and applies it to three parallel worlds, yielding pass rates of 6/15, 6/15, and 0/15 on frontier models with noted qualitative-quantitative asymmetry.
A framework incentivizes inter-coalition collaboration via asymmetric Shapley values and expander graphs to achieve verifiable decentralization in blockchains.
Embedding models fail to capture mathematical equivalence due to lexical bias, but contrastive training on informal-formal pairs improves performance on MELD and retrieval tasks.
LRMs show a large production-evaluation gap on the VAIR dataset with valid answers but invalid reasoning, driven by answer confirmation bias as evidenced by CoT analysis, linear probes, and causal patching.
Proves lattice sphere packings in R^N achieve density at least c N² loglog N 2^{-N} for infinitely many N, improving Klartag's bound via stochastic ellipsoid evolution with cyclotomic symmetries.
Adapts known construction to prove existence of c>0 and large finite A subset R with |AA+A+A| << |A|^{2-c}, plus corollaries for other sum-product expressions.
Optimizes integer multiplicities and a real parameter in Sawin's unit-distance certificate via an evolution strategy to obtain improved explicit lower bounds u(n) > n^{1.0152} and u(n) > n^{1.031}.
AI models discovered the worst-case complexity of the Kaczmarz algorithm for solving linear systems.
An incomplete attempt to show that the O(n^{4/3}) upper bound for unit distances among n points in the plane is not sharp, plus remarks on Szemerédi-Trotter incidences.
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An automated proof that R(B_8,B_10)=37
Proves R(B_8, B_10) = 37 via an AI-assisted short proof with a Lean formalization of the upper bound.