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The Near-Quadratic Elekes-Ronyai Expander Conjecture

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elekes-ronyai-expander-conjectureCombinatoricsposed by Gyorgy Elekes, Lajos Ronyai, 2000recorded: disproved

1 attempt · no person has looked

Statement

The near-quadratic Elekes-Ronyai expander conjecture over R\mathbb{R} predicts that a nonspecial polynomial expands any finite set to near-quadratic size. False: a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of real algebraic integers, has image with a fixed power saving from quadratic size.

Context

The Elekes-Ronyai expansion phenomenon is a pillar of modern incidence geometry and sum-product theory, and the near-quadratic strengthening was the standing expectation.

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  • #1

    Attempt 1

    constructionChatGPT 5.5 Pro, Rethlas with Jihao Liu ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    ChatGPT 5.5 Pro, Rethlas
    people
    Jihao Liu

    The abstract states that the main result was obtained by generative AI, particularly ChatGPT 5.5 Pro and the Rethlas system. The proof builds on a recent OpenAI construction of an infinite tower, so this is an AI result standing on another AI result.

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