🔢 mathematics

On a question of Gowers related to Littlewood's conjecture

This paper provides an explicit construction that answers Gowers' 2009 question by demonstrating the existence of sufficiently many points in the unit cube with large hyperbolic distances, thereby showing that his proposed approach to proving Littlewood's conjecture is insufficient without further refinements.

Original authors: Frederik Broucke, Máté Matolcsi, Szilárd Gy. Révész

Published 2026-07-31

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Frederik Broucke, Máté Matolcsi, Szilárd Gy. Révész

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

Imagine a vast, invisible grid stretching out in every direction, like the floor of a giant, infinite warehouse. In this warehouse, we are playing a game of "hide and seek" with numbers. The game is called Diophantine approximation, and it's all about how well we can approximate messy, irrational numbers (like the square root of 2 or Pi) using simple fractions. The goal is to find fractions that get incredibly close to these tricky numbers without ever actually landing on them.

For decades, mathematicians have been obsessed with a specific puzzle called Littlewood's Conjecture. Think of it as a rule about how close two different numbers can get to being "whole" at the exact same time when you multiply them by a counting number. The rule suggests that no matter which two numbers you pick, if you keep multiplying them by 1, 2, 3, and so on, there will always be a moment where both of them are extremely close to a whole number simultaneously. It's like trying to find a moment when two spinning wheels, one with a weird pattern and one with a different weird pattern, both land on a "zero" mark at the same time. The rule says this will happen, but nobody has been able to prove it for every single pair of numbers yet.

In 2009, a famous mathematician named Timothy Gowers had a clever idea. He thought, "What if we could prove this rule is true by showing that we can't pack too many points into a specific shape without them getting too close to each other?" He imagined a 3D cube and asked if we could scatter a certain number of dots inside it such that the "distance" between any two dots was always large. But here's the twist: the "distance" in this game isn't measured with a ruler. Instead, it's measured by multiplying the differences in the dots' positions across all three dimensions. If you have two dots, and their differences are small in one direction but huge in another, the product might still be big. Gowers wondered if there was a limit to how many dots you could fit before they were forced to be too close in this special "hyperbolic" way.

This brings us to the new paper by Frederik Broucke, Máté Matolcsi, and Szilárd Gy. Révész. They decided to take Gowers' question and build a solution, but not the kind of solution Gowers was hoping for. Instead of finding a way to prove Littlewood's Conjecture using this method, they built a mathematical "trap" that shows the method actually fails.

Here is how they did it: They used a concept from algebraic number theory, which is like a secret code hidden inside numbers. They constructed a very specific, perfectly organized grid of points (called a lattice) using a special type of number field. Imagine this grid as a set of invisible, perfectly aligned pins sticking out of the floor. The authors proved that if you look at the "hyperbolic distance" between any two of these pins, it is never zero unless the pins are the exact same spot. In fact, they showed that you can arrange these points so that they are all far apart according to Gowers' special distance rule.

What does this mean for the big picture? It means that Gowers' proposed approach to proving Littlewood's Conjecture hits a dead end. The authors showed that you can find a huge number of points in the cube that satisfy the "large distance" condition. This is the opposite of what you would need if you wanted to use this method to prove that a counter-example to Littlewood's Conjecture exists. In simple terms, they built a structure that proves the "trap" Gowers set is actually full of holes.

The paper doesn't just say "it doesn't work"; it gives an explicit recipe for building these points. They used a mathematical tool called the Minkowski embedding, which takes numbers from a complex algebraic world and maps them into our regular 3D (or higher-dimensional) space. They showed that for any dimension, you can create a grid where the "product of differences" between any two points is always greater than a specific, tiny number. This proves that the answer to Gowers' question is "Yes, you can find such points," which unfortunately means this specific path cannot be used to solve the original Littlewood mystery.

The authors also connected their findings to a classic problem in Fourier analysis called the Delsarte problem, which is like trying to pack the maximum number of non-overlapping shapes into a box. They showed that their grid construction is related to the best possible way to pack these shapes, giving us a precise number for how many points we can fit. While this doesn't solve Littlewood's Conjecture, it solves Gowers' specific question with a definitive "yes" and a clear, mathematical construction. It's a bit like finding a key that fits a lock perfectly, only to realize the lock opens a door to a room we already knew was empty. The math is solid, the construction is explicit, and the conclusion is clear: this particular strategy needs a major rethink before it can help crack the Littlewood code.

Technical Summary: On a Question of Gowers Related to Littlewood's Conjecture

Problem Statement
The paper addresses a question posed by Timothy Gowers in a 2009 blogpost regarding the potential resolution of Littlewood's conjecture in Diophantine approximation. Littlewood's conjecture asserts that for any real numbers $\alpha$ and $\beta$ , $\liminf_{n\to\infty} n\|n\alpha\|\|n\beta\| = 0$ , where $\|x\|$ denotes the distance from $x$ to the nearest integer.

Gowers proposed a strategy to prove the conjecture by contradiction: if the conjecture were false, there would exist a counterexample $(\alpha, \beta)$ such that the set of points $X_n = \{(k/n, k\alpha \pmod 1, k\beta \pmod 1) : 1 \le k \le n\}$ in the unit cube $[0,1]^3$ would exhibit a specific separation property. Specifically, Gowers asked whether there exists a constant $c > 0$ such that for every positive integer $n$ , one can find $n$ points in $[0,1]^3$ where the "hyperbolic distance" between any two distinct points $(x_1, x_2, x_3)$ and $(y_1, y_2, y_3)$ , defined as $|(x_1-y_1)(x_2-y_2)(x_3-y_3)|$ , is at least $c/n$ .

Methodology
The authors provide an unconditional affirmative answer to Gowers' question (and its generalization to any dimension $d$ ) through an explicit construction based on algebraic number theory.

Lattice Construction: The core of the method involves constructing a full-rank lattice $\Lambda$ in $\mathbb{R}^d$ that intersects the hyperbolic region $H_d = \{x \in \mathbb{R}^d : |x_1 \cdots x_d| < 1\}$ trivially (i.e., $\Lambda \cap H_d = \{0\}$ ).
Algebraic Number Theory: To achieve this, the authors utilize the Minkowski embedding of the ring of integers $\mathcal{O}_K$ of a totally real number field $K$ of degree $d$ . Let $\sigma_1, \dots, \sigma_d$ be the embeddings of $K$ into $\mathbb{R}$ . The lattice is defined as $\Lambda = \{(\sigma_1(\alpha), \dots, \sigma_d(\alpha)) : \alpha \in \mathcal{O}_K\}$ .
Norm Argument: For any non-zero element $\lambda = (\sigma_1(\alpha), \dots, \sigma_d(\alpha)) \in \Lambda$ , the product of its coordinates is the absolute value of the field norm $|N_{K/\mathbb{Q}}(\alpha)|$ . Since $\alpha$ is an algebraic integer, this norm is a non-zero integer, implying $|N_{K/\mathbb{Q}}(\alpha)| \ge 1$ . Consequently, no non-zero lattice point lies within $H_d$ .
Scaling: By scaling the lattice points $\lambda$ by a factor $R^{-1/d}$ and restricting them to a box $[0, R^{1/d}]^d$ , the authors generate a set of points $Y_R$ in the unit cube. The separation property is preserved under this scaling, yielding a hyperbolic distance of at least $1/R$ between distinct points.

Key Contributions and Results

Existence of Separated Points: The paper proves that for any dimension $d$ , there exists a constant $c > 0$ such that for sufficiently large $n$ , one can find $n$ points in $[0,1]^d$ with pairwise hyperbolic distances at least $c/n$ . This confirms Gowers' question in the affirmative.
Explicit Constants: The construction yields an asymptotic constant $c_d = (\min \{\Delta_K : K \text{ is a totally real number field of degree } d\})^{-1/2}$ $c_{d} = (min {Δ_{K} : K is a totally real number field of degree d})^{- 1/2}$ , where $\Delta_K$ $Δ_{K}$ is the discriminant of the field.
- For $d=2$ , using $K=\mathbb{Q}(\sqrt{5})$ , the constant is $c_2 = 1/\sqrt{5}$ .
- For $d=3$ , using the cubic field with discriminant 49, the constant is $c_3 = 1/7$ .
Connection to Delsarte Problem: The authors relate the problem to the Delsarte problem in Fourier analysis. They define $N_d(\varepsilon)$ as the maximal size of a subset in the torus $T^d$ with separation constraints. They show that $N_d(\varepsilon) \asymp 1/\varepsilon$ and establish bounds for the Delsarte constants $D(H_d^\varepsilon)$ and $D'(H_d^\varepsilon)$ using the lattice construction.

Significance and Implications
The primary significance of the paper lies in its negative result regarding the specific approach to Littlewood's conjecture suggested by Gowers. Since the authors have constructed sets of points satisfying the separation condition (which would be a necessary consequence of a counterexample to Littlewood's conjecture) unconditionally, the existence of such sets does not imply the existence of a counterexample.

The authors conclude that the approach of proving Littlewood's conjecture by showing that such separated point sets cannot exist is invalid, as these sets do exist. Therefore, unless further refinements are added to the strategy, this specific line of inquiry cannot succeed in proving Littlewood's conjecture. The paper also notes a curious connection: the constructed sets, which serve as "counterexamples" to the generalized separation problem, consist of elements derived from fixed totally real number field extensions, a setting where Littlewood's conjecture is already known to hold for specific pairs.

Technical Summary: On a Question of Gowers Related to Littlewood's Conjecture

More like this