🔢 mathematics

Pythagorean triples in level sets of completely multiplicative functions

The paper proves that for any finite collection of completely multiplicative functions mapping to the unit circle, there exist Pythagorean triples whose values under these functions are arbitrarily close to 1, utilizing a combination of vanishing averages for aperiodic functions and concentration estimates for pretentious functions to establish this result as a special case of the monochromatic Pythagorean triple conjecture.

Original authors: Guilherme Azevedo, Joel Moreira

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Guilherme Azevedo, Joel Moreira

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

Imagine you are organizing a massive party for the natural numbers (1, 2, 3, 4, and so on). You decide to assign every number a "color" or a "vibe" based on a special rule. In this paper, the authors are looking at a very specific type of rule called a completely multiplicative function.

Think of this rule like a musical instrument. If you play a note for the number 2 and a note for the number 3, the rule says the note for the number 6 (which is 2 times 3) must be the perfect harmony of the 2 and 3 notes combined. These rules can be complex, sometimes sounding chaotic, sometimes sounding very structured.

The Big Question: The Pythagorean Party

The central puzzle the authors are solving is about Pythagorean triples. You know these: three numbers that fit the equation $x^2 + y^2 = z^2$ (like 3, 4, and 5, because $9 + 16 = 25$ ).

For a long time, mathematicians have wondered: If you color the natural numbers in any way you like (using a finite number of colors), will you always find a Pythagorean triple where all three numbers share the same color?

This is a famous open problem. It's like asking: "If I hand out red, blue, and green hats to every person in the world, will I always find three people who fit a specific relationship and are all wearing the same color hat?"

What This Paper Actually Does

The authors, Guilherme Azevedo and Joel Moreira, don't solve the problem for every possible coloring. Instead, they solve it for a very specific, sophisticated type of coloring defined by those "musical" multiplicative rules.

They prove that if you use these specific rules to assign values (which they call "vibes" on the unit circle, a fancy way of saying values that rotate like a clock hand), you can always find a Pythagorean triple where the values for $x$ , $y$ , and $z$ are all extremely close to "1" (which represents a specific, unified state).

In simpler terms: Even with these complex, rotating musical rules, you can always find a set of three numbers that fit the Pythagorean equation and are all singing the same note (or a note very close to it).

How They Did It: The Two-Tool Strategy

The authors had to handle two very different types of these "musical" rules. They used a clever strategy that combined two different tools, like using a sledgehammer for one job and a scalpel for another.

1. The "Chaotic" Tools (Aperiodic Functions)
Some of these rules are like static on a radio or white noise. They are unpredictable and don't repeat.

The Metaphor: Imagine trying to find a pattern in a storm.
The Tool: The authors used a technique called "vanishing averages." It's like saying, "If you listen to this chaotic noise for a long time, the average sound cancels itself out to zero." They proved that for these chaotic rules, the "noise" of the Pythagorean relationship averages out, allowing them to prove the existence of the triple.

2. The "Structured" Tools (Pretentious Functions)
Other rules are very predictable. They "pretend" to be simple, repeating patterns (like a drum beat).

The Metaphor: Imagine a marching band where everyone is perfectly synchronized.
The Tool: For these, the authors used "concentration estimates." This is like saying, "If the band is marching in step, we can predict exactly where they will be." They showed that for these predictable rules, the values cluster tightly together, making it easy to find the triple where everyone is close to the target "note."

The Generalization: More Than Just $x^2 + y^2 = z^2$

The paper also tackles a slightly more general version of the problem: $ax^2 + by^2 = cz^2$ .

The Catch: They can only prove this works if $a$ , $b$ , and $c$ are perfect squares (like 1, 4, 9, 16).
Why? Think of the Pythagorean triple formula as a recipe. The authors found that their recipe only works if the ingredients ( $a, b, c$ ) are "perfect squares." If you try to use other numbers, the recipe breaks down because the mathematical "ingredients" don't mix in the way their tools require.

The "Ergodic" Side Note

The paper briefly mentions a connection to "Ergodic Theory," which is a branch of math dealing with how systems move and mix over time.

The Analogy: Imagine a drop of ink in a glass of water. Ergodic theory asks: "Will the ink eventually spread to every part of the water?"
The authors show that their result is equivalent to saying: "If you have a specific type of mixing machine (a Kronecker system) and you start with a drop of ink, you can always find a moment where the ink forms a specific shape (the Pythagorean triple)." However, they admit this is just a side note and not the main engine of their proof.

Summary

In everyday language:
The authors proved that for a specific, complex class of mathematical rules (completely multiplicative functions), you can always find a Pythagorean triple (or a similar equation with perfect square coefficients) where the three numbers behave almost identically. They did this by splitting the problem into "chaotic" cases (which cancel out) and "orderly" cases (which cluster together), proving that no matter which type of rule you pick, the answer is "yes."

They did not claim to solve the problem for all possible colorings of numbers, nor did they claim to have found a real-world application for this in physics or engineering. They simply closed the door on a specific, difficult mathematical conjecture for a very important class of functions.

Technical Summary: Pythagorean Triples in Level Sets of Completely Multiplicative Functions

Problem Statement
The paper addresses a specific instance of the broader conjecture regarding the partition regularity of the Pythagorean equation $x^2 + y^2 = z^2$ . While Schur's theorem and Rado's theorem established partition regularity for linear equations, the status of the Pythagorean equation under finite colorings of $\mathbb{N}$ remains an open problem (Conjecture 1.1). Specifically, it is unknown whether every finite coloring of the natural numbers contains a monochromatic Pythagorean triple.

The authors investigate this problem through the lens of completely multiplicative functions $f: \mathbb{N} \to S^1$ (where $S^1$ is the unit circle). These functions induce colorings of $\mathbb{N}$ by partitioning $S^1$ into arcs. Previous work had established the existence of Pythagorean triples in level sets of such functions only under the restrictive assumption that the functions are "pretentious" (i.e., they correlate with Dirichlet characters and Archimedean characters). The primary goal of this paper is to remove this restriction and prove the result for any finite collection of completely multiplicative functions, regardless of whether they are pretentious or aperiodic.

Methodology
The proof strategy relies on a dichotomy provided by the Daboussi-Delange lemma, which states that any completely multiplicative function is either pretentious or aperiodic. The authors treat these two cases using distinct analytical tools:

Pretentious Functions: For functions that pretend to be of the form $\chi(n)n^{it}$ (where $\chi$ is a Dirichlet character and $t \in \mathbb{R}$ ), the authors utilize concentration estimates. These estimates, derived from prior work ([5], [13]), allow the control of such functions on specific arithmetic progressions. The authors show that on these progressions, the values of the functions behave predictably, resembling Archimedean characters.
Aperiodic Functions: For functions that do not correlate with any Dirichlet character or Archimedean character, the authors employ Gowers uniformity results. Specifically, they rely on the fact that aperiodic multiplicative functions are Gowers uniform of every order (established in [4]). This implies that statistical averages involving such functions over polynomial configurations vanish.
Parametrization and Averaging: The authors utilize standard polynomial parametrizations for Pythagorean triples (and their generalizations $ax^2 + by^2 = cz^2$ where $a,b,c$ are perfect squares). They define specific averaging schemes over regions $\Delta_N$ in $\mathbb{N}^2$ and use multiplicative Følner sequences to handle the multiplicative structure.
Structural Decomposition: A key technical step involves decomposing the group of multiplicative functions generated by a finite set into a product of "pretentious" components and "aperiodic" components (Lemma 3.2). This allows the authors to reduce the general case to analyzing specific structural lemmas where the behavior of the functions is well-understood.

Key Results
The paper establishes the following main theorems:

Theorem 1.2: For any finite collection of completely multiplicative functions $f_1, \dots, f_d$ and any open set $I \subset S^1$ containing 1, there exists a Pythagorean triple $(x, y, z)$ such that $f_i(x), f_i(y), f_i(z) \in I$ for all $i$ . This confirms the conjecture for colorings induced by multiplicative functions without the pretentiousness assumption.
Theorem 1.4: The result is extended to the more general equation $ax^2 + by^2 = cz^2$ , provided that $a, b, c$ are perfect squares and satisfy either $a=c$ or $a+b=c$ (satisfying Rado's condition).
Theorem 1.12 (Jointly Pythagorean): The authors introduce the concept of "jointly Pythagorean" functions. They prove that any finite collection of completely multiplicative functions is jointly Pythagorean with respect to the specified equations. This is a stronger statement than the existence of a single triple; it implies that the functions simultaneously map the components of the triple into the target neighborhood of 1 with positive density.

Significance and Claims
The paper claims to provide a new special case of the conjecture that any finite coloring of $\mathbb{N}$ has a monochromatic Pythagorean triple. By removing the "pretentious" condition, the authors generalize previous results (specifically [7]) to cover the full class of completely multiplicative functions.

The authors note that their proof combines two distinct regimes:

Vanishing averages for aperiodic functions (using Gowers uniformity).
Concentration estimates for pretentious functions.

A limitation acknowledged by the authors is that the reliance on these specific tools restricts the result to equations where the parametric solutions involve only one irreducible polynomial. This is why the coefficients $a, b, c$ must be perfect squares. The paper does not claim to solve the general partition regularity of $x^2 + y^2 = z^2$ for arbitrary colorings, nor does it claim to resolve the case for three or more colors in the general setting, but it significantly advances the understanding of the problem within the framework of multiplicative functions.

The work also provides an ergodic interpretation (Theorem 1.7) of the main result, linking it to double recurrence statements in multiplicative measure-preserving systems, specifically for Kronecker systems. This suggests a connection between the combinatorial properties of multiplicative functions and ergodic theory, although the authors note that lifting this result to general systems (where the Kronecker factor is trivial) remains an open challenge.