A counterexample to the near-quadratic Elekes--Rónyai expander conjecture over $\mathbb R$

Abstract

We disprove the near-quadratic Elekes--Rónyai expander conjecture over $\mathbb R$. The counterexample is a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of real algebraic integers on which its image has a fixed power saving from quadratic size. The main result of this paper is obtained by generative AI, particularly ChatGPT 5.5 Pro and the Rethlas system. The proof relies on a recent construction by OpenAI of an infinite tower of number fields.