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A counterexample to the near-quadratic Elekes–Rónyai expander conjecture over ℝ

Jihao Liu

For f(x,y)=(x−y)²+Mx with a fixed suitable M, there are arbitrarily large finite real sets A with |f(A,A)|≤|A|^(2−c) for some c>0, although f has neither additive nor multiplicative special form.

Sources for this record

What was already known

The conjecture predicted nearly quadratic image growth for every nonspecial real polynomial. Pohoata independently announced a different disproof before Liu’s public paper.

AI involvement AI-led

Liu credits Rethlas with the retained construction and most of the paper, with ChatGPT suggesting the arithmetic direction and a different polynomial. Liu wrote the postscript and clarified a lemma after correspondence.

The author says Rethlas generated the retained fixed nonspecial polynomial construction and nearly all of the paper, with ChatGPT suggesting a different test polynomial and the arithmetic direction. Human steering and existing inputs remain explicit.

Importance Notable

Provides a distinct fixed-polynomial counterexample with a power saving over quadratic image growth.

Limitations

The author clarified a proof lemma after expert feedback, but no completed check of the whole argument is documented. The model may have drawn on Pohoata’s earlier page; its precise independence is unresolved. Pohoata’s different proof does not check this argument.

Technical verification Partial

The postscript reports outside expert feedback and corrects the presentation of Lemma 2.1, a part of the proof. It does not identify a completed rigorous human or machine check of the entire theorem. The author says two signed lemmas were checked, but the checkers and scope are not specified.

Independent checking

Liu clarified the box-estimate lemma after expert feedback; no completed check of the whole theorem is documented.

Checker
Jihao Liu, informed by other experts and correspondence with Cosmin Pohoata
Scope
The box-estimate Lemma 2.1 used in the counterexample proof
Basis
Reported by the linked checker
Evidence source

Human understanding Few

The AI-written paper explains the mechanism, and Liu clarifies a box-estimate lemma, but no named human explains the full central AI construction. Few is the evidence default for the assessed proof, not a statement of inability.

Correction and publication history