Near-quadratic Elekes–Rónyai expansion outside special forms
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Must a bivariate polynomial with neither additive nor multiplicative special form have nearly quadratic image growth on A×A for large finite real A? The specific target is f(x,y)=(x−y)²+Mx for one fixed suitable M.
Why it matters and what counts as progress
Why it matters: A fixed nonspecial polynomial with subquadratic image growth would limit a broad expansion prediction beyond additive and multiplicative forms.
What counts as progress: Construct arbitrarily large sets with a fixed power saving in image size, or prove stronger expansion for an explicit polynomial class.
How the problem developed
György Elekes and Lajos Rónyai Established superlinear expansion for nonspecial bivariate polynomials, motivating later questions about whether expansion must be close to quadratic.
- 15 Jun 2026 · Counterexample [2]
Jihao Liu; the paper credits generative AI, particularly ChatGPT 5.5 Pro and Rethlas, with the main result Disproved the near-quadratic strengthening over ℝ using a fixed nonspecial quadratic polynomial and arbitrarily large sets of real algebraic integers with a fixed power saving from quadratic image size.
AI contribution
A counterexample to the near-quadratic Elekes–Rónyai expander conjecture over ℝ
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.
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AI-led counterexample to near-quadratic Elekes–Rónyai expansion