Split primes and the Elekes-Rónyai problem

Abstract

There exist an absolute constant $c>0$ and arbitrarily large finite sets $A\subset \mathbb{R}$ with $$\left|\left\{x+y+(x-y{)}^2:x,y\in A\right\}\right|\le |A{|}^{2-c}.$$ Since $x+y+(x-y{)}^2\in \mathbb{R}[x,y]$ is a polynomial which is neither additive nor multiplicative, this provides a counterexample for the Elekes-Rónyai problem. The proof combines two amplifications of the same local congruence defect: horizontal amplification over squarefree products of rational primes, and vertical amplification through bounded root-discriminant towers in which those primes split completely. In this way a fixed local density defect becomes macroscopic, producing a power saving. This phenomenon also suggests a broader mechanism for producing similar extremal constructions throughout combinatorics and number theory.

14 pages, new Section 5 added