🔢 mathematics

The sum-product conjecture is false for real numbers

This paper refutes the sum-product conjecture for real numbers and related settings by constructing arbitrarily large sets with simultaneously small sum and product sets, thereby disproving the conjecture and the many sums and products conjecture while establishing new bounds for solutions to linear and unit equations.

Original authors: Thomas F Bloom, Will Sawin, Carl Schildkraut, Dmitrii Zhelezov

Published 2026-05-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Thomas F Bloom, Will Sawin, Carl Schildkraut, Dmitrii Zhelezov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: Breaking a 50-Year-Old Rule

Imagine you have a bag of numbers. You can do two things with them:

Add them together (create a "Sum Set").
Multiply them together (create a "Product Set").

For decades, mathematicians believed in a rule called the Sum-Product Conjecture. The rule was simple: You can't have both.

The intuition was that if your numbers are "clumped" together so that adding them doesn't create many new numbers (like a straight line of numbers), then multiplying them should explode into a huge variety of new numbers. Conversely, if multiplying them keeps them small (like a geometric progression), then adding them should create a massive explosion of new numbers.

The conjecture stated that for any large set of real numbers, at least one of these operations must create a huge number of results. It was thought to be impossible to have a set where both addition and multiplication stay small.

This paper says: "Not so fast."

The authors have built a mathematical "magic trick" that disproves this rule for real numbers. They constructed specific, massive sets of numbers where both the sum set and the product set remain surprisingly small. They didn't just break the rule; they broke it with a massive, arbitrarily large set of numbers.

The Magic Trick: How They Did It

To understand their construction, imagine you are building a city.

1. The Old Way (The Balog-Wooley Example)

Previously, mathematicians tried to make a small set by mixing two types of neighborhoods:

The Geometric Street: A street where houses are spaced out exponentially (1, 10, 100, 1000...). If you multiply these, you stay on the street. But if you add them, the gaps are so huge that the sums spread out everywhere.
The Arithmetic Street: A street where houses are evenly spaced (1, 2, 3, 4...). If you add them, they stay in a tight block. But if you multiply them, the numbers explode.

The old trick was to mix these streets. But the "Geometric Street" was too sparse. It was like having a few houses on a massive highway. When you added them, the result was still too big to break the rule.

2. The New Way (The High-Dimensional Lattice)

The authors realized they needed a "Geometric Street" that was dense (packed with houses) but still behaved nicely when multiplied. They found this by moving from a flat 2D map to a high-dimensional hyper-city.

The City (Number Fields): Instead of working with normal integers, they built their numbers inside a special "universe" called a totally real number field. Think of this as a city with $d$ different dimensions (where $d$ is a very large number).
The Additive Grid (The Box): They picked a block of numbers that looks like a perfect, tight cube in this high-dimensional space. When you add numbers from this cube, the result is just a slightly larger cube. It stays compact.
The Multiplicative Grid (The Units): They also picked a special group of numbers called "units." In this high-dimensional world, these units behave like a dense geometric progression. When you multiply them, they stay within a tight, predictable shape.

3. The Secret Sauce: The "Direct Product"

The genius move was combining these two grids. They took every number from the "Additive Cube" and multiplied it by every number from the "Multiplicative Grid."

Why it works: Because the "Additive Cube" is so tightly packed and the "Multiplicative Grid" is so dense, the resulting set is huge (arbitrarily large).
The Result:
- When you add two numbers from this new set, the "Additive Cube" part keeps the result from exploding.
- When you multiply two numbers, the "Multiplicative Grid" part keeps the result from exploding.

It's like having a city where the streets are so perfectly designed that no matter how you walk (add) or how you drive (multiply), you never leave a small, manageable neighborhood.

Why This Matters (According to the Paper)

The paper doesn't just say "we broke the rule." It shows how to break it in different mathematical worlds:

Real Numbers: They proved the rule is false for standard real numbers (the numbers we use in daily life).
P-adic Numbers: They showed the same trick works in a different type of number system used in advanced physics and cryptography.
Finite Fields: They showed that even in "clock arithmetic" (where numbers wrap around), you can find sets that break the rule.
Function Fields: They extended this to fields of formal power series (used in coding theory).

The "AI" Note

The authors mention that they were inspired by a recent AI-generated counterexample to a different math problem (the "unit distance conjecture"). While they used an AI (GPT-5.5 Pro) as a "sounding board" to help simplify one specific lemma (a small proof step), the core ideas, the main construction, and the entire proof were generated by humans. The AI was just a helpful assistant, not the architect.

Summary

For 50 years, mathematicians thought you couldn't have a set of numbers that is "small" under both addition and multiplication. This paper says: "We built one."

They did it by constructing a massive, high-dimensional mathematical structure where the rules of addition and multiplication are perfectly balanced to keep the results small. They proved that the old rule was wrong, not just for a tiny, weird exception, but for sets of numbers that can be as large as you want.

Technical Summary: The Sum-Product Conjecture is False for Real Numbers

Problem Statement
The paper addresses the Erdős–Szemerédi sum-product conjecture, which posits that for any finite set $A$ in a ring (specifically $\mathbb{Z}$ , $\mathbb{R}$ , or $\mathbb{C}$ ), at least one of the sum set $A+A = \{a+b : a,b \in A\}$ or the product set $AA = \{ab : a,b \in A\}$ must grow near-maximally. Formally, the conjecture asserts:
$\max(|A+A|, |AA|) \ge |A|^{2-o(1)}$
where the $o(1)$ term vanishes as $|A| \to \infty$ . While the best known lower bounds for subsets of $\mathbb{R}$ are of the form $|A|^{4/3+c}$ , the conjecture remains open. The authors investigate whether this conjecture holds true over the real numbers.

Methodology
The authors construct explicit counterexamples using algebraic number theory, specifically leveraging totally real number fields of large degree. The core strategy involves a high-dimensional generalization of the Balog–Wooley example, replacing the standard geometric progression and arithmetic progression with structured subsets of the ring of integers $\mathcal{O}_K$ of a number field $K$ .

The construction relies on two distinct lattice structures within $\mathcal{O}_K$ :

The Additive Lattice: A "box" $P$ of algebraic integers where all real embeddings $\sigma_i(p)$ lie in a short interval $[X-cX, X+cX]$. This ensures the additive structure is confined to a small region.
The Multiplicative Lattice: A set $G$ of units $u \in \mathcal{O}_K^\times$ where the logarithms of their embeddings $|\log|\sigma_i(u)||$ are bounded by a constant $Y$ . This creates a set with small product growth (small doubling).

The counterexample set is defined as the product $A = GP = \{up : u \in G, p \in P\}$ . The authors ensure this product is direct (i.e., $|A| = |G||P|$ ) by choosing parameters such that the ratio of any two elements in $P$ cannot be a non-trivial unit, a property guaranteed by Schinzel's results on the separation of units.

The existence of suitable number fields is guaranteed by Martinet's construction of towers of totally real fields with bounded root discriminant ( $\Delta_K \le C^d$ ) and controlled regulators ( $R_K \le \Delta_K$ ). These bounds ensure that the lattice volumes (and thus the sizes of $P$ and $G$ ) grow exponentially with the degree $d$ , allowing the construction of arbitrarily large sets $A$ .

Key Contributions and Results

Disproof of the Sum-Product Conjecture over $\mathbb{R}$ :
The primary result (Theorem 1.1) establishes the existence of an absolute constant $c > 0$ such that there are arbitrarily large finite sets $A \subset \mathbb{R}$ satisfying:
$\max(|A+A|, |AA|) \le |A|^{2-c}$
This directly contradicts the conjecture. The construction yields sets where the sum set is small ( $|A+A| \le |A|^{1+\epsilon}$ ) and the product set exhibits power-saving ( $|AA| \le |A|^{2-c\epsilon}$ ).
Disproof of the "Many Sums and Products" Conjecture:
The authors extend their construction to disprove the stronger conjecture by Erdős regarding $k$ -fold sums and products. For any fixed $k \ge 3$ , they construct arbitrarily large $A \subset \mathbb{R}$ such that:
$\max(|kA|, |A^{(k)}|) \le |A|^{C \frac{\log k}{\log \log k}}$
This dependence on $k$ is noted as likely optimal for fixed $k$ as $|A| \to \infty$ .
New Lower Bounds for Linear Equations in Multiplicative Groups:
The construction provides new lower bounds for the number of solutions to linear equations $x_1 + \dots + x_k = 1$ with $x_i$ in a multiplicative group $\Gamma$ .
- For subgroups of $\mathbb{R}^\times$ of rank $d$ , the number of non-degenerate solutions is shown to be at least $\exp((C^{-1}k \log k)d)$ , demonstrating that the linear dependence on the rank $d$ in the exponent of the Evertse–Schlickewei–Schmidt upper bound is best possible.
- For the unit equation in number fields, the authors show that for sufficiently large $k$ , there exist fields of degree $d$ with at least $C^d$ solutions, disproving conjectures of sub-exponential growth for $k \ge 3$ .
Variants in Other Settings:
The methodology is adapted to produce counterexamples in:
- $p$ -adic numbers ( $\mathbb{Q}_p$ ): Using fields where $p$ splits completely.
- Finite fields ( $\mathbb{F}_p$ ): By reducing the algebraic integer construction modulo $p$ , yielding sets of size $p^c < |A| < p^{1/2}$ with small sum-product growth.
- Function fields ( $\mathbb{F}_q((t))$ ): Using line bundles on algebraic curves with many rational points (Drinfeld–Vlăduț–Zink sequences) to construct counterexamples in positive characteristic.

Significance and Claims
The paper claims to definitively disprove the sum-product conjecture for real numbers, a problem that has been central to additive combinatorics for decades. The authors emphasize that while the conjecture may still hold for sets in bounded-degree number fields (such as $\mathbb{Z}$ ), it fails in the general setting of $\mathbb{R}$ due to the availability of high-degree algebraic structures.

The authors note that their proof relies on the existence of number fields with specific arithmetic properties (bounded discriminant and regulator), which are non-trivial to construct but well-established in algebraic number theory. They explicitly state that the final proof, including the main ideas, was almost entirely human-generated, though they utilized AI as a "sounding board" in early stages and for a specific elementary lemma (Lemma 3.4) that simplified a more complex result.

The work also provides new quantitative insights into the limits of sum-product phenomena, showing that the "power saving" in product sets can be achieved simultaneously with small sum sets, provided the underlying algebraic structure is sufficiently rich (high degree). The paper concludes by offering explicit, albeit small, values for the constant $c$ (e.g., $c \ge 0.00000087$ ) and demonstrating that similar phenomena occur across various algebraic settings, including $p$ -adics and function fields.