The M öbius Disjointness Conjecture on infinite-dimensional torus

This paper proves that the Möbius Disjointness Conjecture holds for a specific class of irregular, distal flows on the infinite-dimensional torus defined by a skew-product transformation with $C^{1+\varepsilon}$-smooth coefficients.

The Gist

Here is an explanation of the paper "The Möbius Disjointness Conjecture on Infinite-Dimensional Torus," translated into everyday language with creative analogies.

The Big Picture: Chaos vs. Order

Imagine you have two very different worlds:

1. The World of Prime Numbers: This is the realm of the Möbius function. Think of prime numbers as the "atoms" of mathematics. They are unpredictable, scattered, and seem to have no pattern. The Möbius function is a tool mathematicians use to track these primes. It acts like a coin flip: sometimes it's +1, sometimes -1, and sometimes 0. It represents pure, random noise.
2. The World of Motion: This is the realm of Dynamical Systems. Imagine a machine with gears, wheels, and spinning parts. Even if the machine looks complicated, it follows strict rules. If you know where a gear starts, you can predict exactly where it will be later. This represents "order" or "determinism."

The Big Question (Sarnak's Conjecture):
In 2009, mathematician Peter Sarnak asked a profound question: If you mix the "random noise" of prime numbers with the "strict order" of a mechanical machine, do they cancel each other out?

His guess (the Möbius Disjointness Conjecture) is YES. He believes that if a machine is "simple enough" (mathematically speaking, having zero entropy), the random fluctuations of prime numbers will average out to zero when interacting with the machine. The primes and the machine are "disjoint"—they don't talk to each other.

The New Discovery: The Infinite Spinning Top

This paper by Qingyang Liu, Jing Ma, and Hongbo Wang proves that Sarnak's guess is true for a very specific, incredibly complex machine: the Infinite-Dimensional Torus.

What is this machine?

Imagine a standard spinning top (a circle). Now, imagine a second top attached to the first one, a third attached to the second, and so on, forever.

• The Base: The first top spins at a steady speed ($\alpha$).
• The Chain Reaction: The second top doesn't just spin on its own; its speed depends on where the first top is. The third top depends on the second, and so on.
• The Twist: This isn't just a simple chain. The paper looks at a version where the connection between the tops is "smooth" but "twisted" (mathematically, a $C^{1+\epsilon}$ smooth function).

This creates an Infinite-Dimensional Torus ($T^\omega$). It's a shape with infinite directions you can move in, all spinning together in a coordinated, yet complex, dance.

The Problem

Proving that primes and this infinite machine don't "interact" is hard because the machine is infinite. Usually, when things get infinite, math breaks down or becomes too messy to solve.

The Solution: Two Ways to Prove It

The authors didn't just guess; they built two different "bridges" to prove the machine and the primes are disjoint.

Bridge 1: The "Stiffness" Test (Polynomial Rigidity)

• The Analogy: Imagine a very stiff, rigid metal rod. If you push it slightly, it barely bends. If you push it again and again at specific intervals, it returns to its original shape almost perfectly.
• The Math: The authors showed that this infinite machine has a property called "Polynomial Rate of Rigidity." This means that if you let the machine run for a long time, it keeps coming back to positions that look almost exactly like where it started.
• The Result: Because the machine is so "stiff" and predictable in its return, the random noise of the primes gets washed out. The primes can't "lock in" to the machine's rhythm because the machine is too rigid.

Bridge 2: The "Complexity" Test (Sub-polynomial Measure Complexity)

• The Analogy: Imagine trying to describe the weather in a city. If the weather is chaotic (like a storm), you need a massive amount of data (high complexity) to predict it. If the weather is simple (like a calm day), you need very little data (low complexity).
• The Math: The authors measured how "complex" the machine's movement is. They proved that even though the machine has infinite parts, its movement is actually "simple" in a specific mathematical sense (sub-polynomial complexity). It doesn't generate enough new, chaotic information to hide from the primes.
• The Result: Since the machine isn't truly chaotic, the primes (which are random) have nothing to grab onto. They pass right through without interacting.

Why Does This Matter?

1. It's a Victory for "Simple" Chaos: This machine looks incredibly complicated (infinite dimensions!), but the paper proves it's actually "simple" enough that the rules of prime numbers still apply. It extends our understanding of how order and randomness interact.
2. No Special Rules Needed: Previous proofs for similar machines required the spinning speeds to be "perfectly irrational" (a very specific, rare type of number). This paper proves it works for any real number speed, making the result much more robust.
3. Two Paths to the Same Truth: By proving it using two completely different mathematical tools (Rigidity and Complexity), the authors made the result much harder to dispute. It's like proving a bridge is safe by testing it with both a heavy truck and a strong wind.

The Bottom Line

The authors took a wild, infinite, twisting machine and showed that it is "boring" enough that the mysterious, random behavior of prime numbers simply ignores it. They proved that even in an infinite world, the rules of number theory hold firm against the chaos of motion.

Technical Summary

Here is a detailed technical summary of the paper "The Möbius Disjointness Conjecture on Infinite-Dimensional Torus" by Qingyang Liu, Jing Ma, and Hongbo Wang.

1. Problem Statement

The paper addresses Sarnak's Möbius Disjointness Conjecture within the context of infinite-dimensional dynamical systems.

• The Conjecture: Sarnak's conjecture posits that the Möbius function $\mu(n)$ is linearly disjoint from any zero-entropy topological dynamical system $(X, T)$. Mathematically, for any continuous function $f \in C(X)$ and any point $x \in X$:
  $$\lim_{N \to \infty} \frac{1}{N} \sum_{n \le N} \mu(n) f(T^n x) = 0$$
• The System: The authors study a specific class of flows on the infinite-dimensional torus $\mathbb{T}^\omega = \prod_{k=1}^\infty \mathbb{T}$ (where $\mathbb{T} = \mathbb{R}/\mathbb{Z}$). The transformation $T: \mathbb{T}^\omega \to \mathbb{T}^\omega$ is defined as a skew product:
  $$T(x_1, x_2, \dots, x_k, \dots) = (x_1 + \alpha, x_2 + h(x_1), \dots, x_k + h(x_1 + (k-2)\beta), \dots)$$
  where:
  • $\alpha \in \mathbb{R}$ and $\beta \in \mathbb{R} \setminus \mathbb{Q}$ (irrational).
  • $h: \mathbb{R} \to \mathbb{R}$ is a $1$-periodic function of class$C^{1+\varepsilon}$(smoothness slightly better than$C^1$).
• Significance of the System: These flows are distal (points remain separated under iteration) but are irregular in the sense that their Birkhoff averages do not exist for all points. While the conjecture has been verified for finite-dimensional tori (e.g., the 2-torus) and specific "Furstenberg irregular flows" on infinite-dimensional tori, this paper aims to prove it for a much broader class of skew products without restrictive arithmetic conditions on the coefficients of $h$.

2. Methodology

The authors employ a dual approach, splitting the proof based on the arithmetic nature of $\alpha$ and utilizing two distinct dynamical criteria established in recent literature.

A. Case 1: $\alpha$ is Rational

• Reduction: When $\alpha$ is rational, the dynamics on the base circle become periodic.
• Technique: The problem reduces to estimating exponential sums involving the Möbius function.
• Tool: The authors apply a classical theorem by Davenport regarding exponential sums with the Möbius function against polynomials. This allows them to bound the sum $\sum \mu(n) e(P(n))$ effectively, proving disjointness directly via number-theoretic estimates.

B. Case 2: $\alpha$ is Irrational

For irrational $\alpha$, the authors utilize two independent dynamical criteria that imply Möbius disjointness. They prove that the system satisfies these criteria under different smoothness assumptions on $h$.

Criterion 1: Polynomial Rate of Rigidity (PR Rigidity)

• Source: Kanigowski, Lemanczyk, and Radziwill [10].
• Concept: A system is PR rigid if there exists a sequence of times $\{r_n\}$ such that the system returns close to the identity at a polynomial rate for a dense set of functions.
• Application: The authors prove Theorem 3.1, showing that for $h \in C^{1+\varepsilon}$, the system $(\mathbb{T}^\omega, T)$ has PR rigidity for all invariant measures.
• Proof Strategy:
  1. Define a metric on $\mathbb{T}^\omega$ and analyze the distance $d(T^{r_n}x, x)$.
  2. Utilize the continued fraction expansion of $\alpha$ to select a sequence of denominators $q_n$.
  3. Construct a sequence $r_n$ based on $q_n$ to minimize the discrepancy.
  4. Use Parseval's identity and estimates on exponential sums (Lemmas 4.1 and 4.2) to bound the $L^2$ norm of the difference $f \circ T^{r_n} - f$.
  5. Show that the error decays polynomially, satisfying the PR rigidity condition.

Criterion 2: Sub-Polynomial Measure Complexity

• Source: Huang, Wang, and Ye [9].
• Concept: The measure complexity $s_n(d, \nu, \varepsilon)$ counts the minimum number of $\varepsilon$-balls (under the dynamical metric $\bar{d}_n$) needed to cover most of the space. If this growth is slower than any polynomial $n^\tau$, the system has sub-polynomial complexity.
• Application: The authors prove Theorem 3.2, showing that for $h \in C^\infty$ (smooth), the system has sub-polynomial measure complexity.
• Proof Strategy:
  1. Conjugacy: They construct a measurable isomorphism (conjugacy) between the original flow $T$ and a simpler flow $S$ (or $\tilde{S}$) by solving a cohomological equation involving the function $h$.
  2. Case Distinction: They distinguish between cases where a specific set $M$ (related to the denominators of $\alpha$ and Fourier coefficients of $h$) is finite or infinite.
     • If $M$ is finite, the system is conjugate to a rotation (discrete spectrum), which trivially has bounded complexity.
     • If $M$ is infinite, they use the rapid decay of Fourier coefficients of $h$ and the properties of the set $E_\tau$ (indices where denominators grow super-exponentially) to construct a covering of the space with a sub-polynomial number of balls.
  3. Conclusion: Since the conjugate system has sub-polynomial complexity, the original system does as well.

3. Key Contributions and Results

• Main Theorem (Theorem 1.1): The Möbius Disjointness Conjecture holds for the infinite-dimensional skew product flow defined in (1.2) for all $\alpha \in \mathbb{R}$ and all $h \in C^{1+\varepsilon}$.
• Generalization of Previous Work:
  • Extends the results of Liu and Sarnak (2015) from the 2-torus to infinite dimensions.
  • Extends the work of Liu (2024) on "Furstenberg's irregular flows." The previous work required $h$ to satisfy specific technical conditions related to the Fourier coefficients (Eq. 1.4 in the paper). This paper removes those restrictions, allowing for general $C^{1+\varepsilon}$ functions.
  • Works for all $\alpha$, whereas previous infinite-dimensional results often required Diophantine conditions or specific growth rates for the denominators of $\alpha$.
• Dual Proof Mechanism: The paper provides two independent proofs for the irrational $\alpha$ case:
  1. Via PR rigidity (requiring only $C^{1+\varepsilon}$ smoothness).
  2. Via sub-polynomial measure complexity (requiring $C^\infty$ smoothness).
     This demonstrates the robustness of the result and highlights the interplay between different dynamical criteria.

4. Significance

• Bridging Number Theory and Dynamics: The paper reinforces the deep connection between the distribution of prime numbers (via the Möbius function) and the ergodic properties of infinite-dimensional systems.
• Infinite-Dimensional Dynamics: It represents a significant step forward in understanding the ergodic theory of infinite-dimensional tori, a setting that is notoriously difficult due to the lack of compactness in the standard sense (though $\mathbb{T}^\omega$ is compact, the dynamics are complex).
• Relaxing Regularity Conditions: By proving the conjecture for $C^{1+\varepsilon}$ functions (the lowest regularity threshold currently known for such skew products in finite dimensions), the authors push the boundary of what is known about the minimal smoothness required for Möbius disjointness.
• Methodological Advancement: The successful application of PR rigidity and measure complexity criteria to infinite-dimensional skew products provides a new toolkit for researchers tackling similar problems in higher-dimensional or infinite-dimensional settings.

In summary, this paper confirms that the "randomness" of the Möbius function is strong enough to be orthogonal to the deterministic, yet irregular, dynamics of a broad class of infinite-dimensional skew products, regardless of the arithmetic nature of the rotation parameter $\alpha$.