Split primes and the Elekes-Rónyai problem

Abstract: There exist an absolute constant $c&gt;0$ and arbitrarily large finite sets $A\subset \mathbb{R}$ with $$\left| \left{x+y+(x-y)^2:\ x, y \in A\right}\right| \le|A|<sup>{2-c}.$$ 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 paper demonstrates an explicit counterexample to the Elekes-Rónyai conjecture by constructing a non-special polynomial with subquadratic image growth.
• It leverages combinatorial large sieve techniques and arithmetic properties of split primes in towers of number fields to restrict the image size.
• The construction yields sets with small additive doubling and introduces novel lattice methods, challenging current bounds in polynomial expansion theory.

Counterexamples to the Elekes-Rónyai Problem via Split Primes

Introduction

The Elekes-Rónyai problem concerns image set growth for bivariate polynomials in arithmetic combinatorics. Given $f \in \mathbb{R}[x, y]$ and finite subsets $A, B \subset \mathbb{R}$ of size $n$, the focus is on bounding $|f(A, B)|$. For so-called "special" polynomials—additive or multiplicative forms—$|f(A,B)|$ can be kept $O(n)$. Elekes and Rónyai established that, unless $f$ is special, $|f(A, B)|/n \to \infty$ as $n\to\infty$. Elekes further conjectured that for every non-special $f$ and any $A, B \subset \mathbb{R}$0, there exists $A, B \subset \mathbb{R}$1 such that $A, B \subset \mathbb{R}$2, i.e., nearly quadratic lower bounds.

The paper "Split primes and the Elekes-Rónyai problem" (2606.13619) disproves this conjecture by producing an explicit non-special polynomial and infinite families of large sets $A, B \subset \mathbb{R}$3 with cardinality $A, B \subset \mathbb{R}$4 for which $A, B \subset \mathbb{R}$5 for some absolute constant $A, B \subset \mathbb{R}$6. The construction leverages combinatorial large sieve techniques and deep arithmetic input regarding split primes in towers of totally real fields.

The Elekes-Rónyai Framework and Existing Results

The image size lower bound for non-special polynomials is a central problem at the intersection of additive combinatorics and incidence geometry. Quantitative progress on the conjectured nearly quadratic bound ($A, B \subset \mathbb{R}$7) has been slow. The best unconditional lower bound remains $A, B \subset \mathbb{R}$8, due to Solymosi and Zahl (2024). All known progress relied on the incompatibility of additive and multiplicative structures.

Construction of the Counterexample

Polynomial Choice and its Non-Specialness

The chosen polynomial is $A, B \subset \mathbb{R}$9, which is not of additive nor multiplicative Elekes-Rónyai special type. An explicit verification is provided in the paper to show that $n$0 cannot be written as $n$1 nor as $n$2 for $n$3 univariate (see (2606.13619), Claim).

Reduction to Arithmetic Restrictions via Split Primes

The construction relies on a polynomial $n$4, with $n$5 a product of odd rational primes $n$6 chosen to all split completely in a tower of totally real number fields $n$7, with rapidly growing degree $n$8. For each $n$9 and each embedding $|f(A, B)|$0, reduction modulo the corresponding split prime ideal $|f(A, B)|$1 kills the linear part, compelling $|f(A, B)|$2 to be a square in the finite field $|f(A, B)|$3. As a consequence, the image of $|f(A, B)|$4 is forced into roughly $|f(A, B)|$5 out of $|f(A, B)|$6 residue classes in each field.

This "local bottleneck" is globalized across all split primes and embeddings, resulting in an exponential saving: the set of possible image residues has relative density $|f(A, B)|$7 with $|f(A, B)|$8 (explicitly, $|f(A, B)|$9).

Lattice and Geometry-of-Numbers Techniques

Sets $|f(A,B)|$0 are obtained by projecting "symmetric Minkowski boxes" in the ring of integers $|f(A,B)|$1 under a real embedding, yielding $|f(A,B)|$2 for box radius $|f(A,B)|$3. Key geometry-of-numbers estimates control $|f(A,B)|$4, $|f(A,B)|$5, and ensure small additive doubling ($|f(A,B)|$6 for any $|f(A,B)|$7). The Chinese Remainder Theorem and the bounded root discriminant tower guarantee that the image compresses exponentially in $|f(A,B)|$8.

Quantitative Image Set Bound

For each large $|f(A,B)|$9 (i.e., large $O(n)$0), this machinery provides

$O(n)$1

with $O(n)$2 a constant determined by the residue density and the rate at which $O(n)$3 grows with $O(n)$4 (see (2606.13619), Section 4). The construction is uniform in the sense that the polynomial $O(n)$5 is fixed, thus providing a genuine counterexample to the Elekes conjecture.

Theoretical and Practical Implications

The principal implication is that, contrary to the widespread conjecture, non-special bivariate polynomials can have genuinely subquadratic image growth. This severely restricts the possible scope of polynomial expansion theorems over the real numbers, and compels a total revision of the quantitative theory at the interface of arithmetic combinatorics and incidence geometry.

The proof techniques introduce a new use of combinatorial large sieve methods, further developed in joint work by Croot, Mao, Pohoata, Sheffer, and Yip, and demonstrate the utility of explicit towers of number fields with bounded root discriminant and plentiful split primes—a tool previously pivotal in the recent disproofs of the Erdős unit distance and sum-product conjectures.

More broadly, the approach suggests that similar techniques may yield new insight into other expansion and anti-concentration conjectures in higher dimensions or for other classes of polynomials or rational functions. The fine structure of lattice points and local-to-global phenomena in arithmetic combinatorics, as harnessed here, is likely to inform future work.

Numerical and Structural Highlights

• There is an explicit, fixed, non-special $O(n)$6 and absolute $O(n)$7 with $O(n)$8 for arbitrarily large $O(n)$9.
• The constructed sets $f$0 can simultaneously satisfy $f$1.
• The residue bottleneck is exponential in the number field degree, producing a power saving in image size.

Conclusion

"Split primes and the Elekes-Rónyai problem" (2606.13619) provides the first explicit counterexamples to the anticipated quadratic image set lower bounds for non-special polynomials over the reals, refuting the Elekes conjecture. The work integrates combinatorial large sieve, advances in arithmetic geometry, and geometric lattice methods, with implications for all future research on polynomial expansion, distinct distances, and related combinatorial geometry problems. These results indicate a need for revised conjectures accounting for the phenomena driven by arithmetic input and local-global constraints in higher-degree settings.