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Möbius Disjointness Conjecture for a skew product on a circle and the Heisenberg nilmanifold

Yuk-Kam Lau, Jing Ma

TL;DR

The paper proves Sarnak's Möbius disjointness conjecture for a skew product on the circle coupled with the 3-dimensional Heisenberg nilmanifold, removing a previous symmetry condition. The authors establish polynomial-rate rigidity for irrational rotation parameters and, under smoothness assumptions, derive sub-polynomial measure complexity, enabling Möbius disjointness via PR rigidity and HWY-type criteria. The approach blends continued-fraction analysis, harmonic analysis on the Heisenberg group, and a careful decomposition into resonant and non-resonant parts to control double exponential sums arising from the skew product. These results extend Möbius disjointness to a broader class of distal skew products on non-abelian nilmanifolds, contributing substantial evidence toward Sarnak's conjecture in a non-commutative setting.

Abstract

We establish Sarnak's conjecture on Möbius disjointness for the dynamical system of a skew product on a circle and the three-dimensional Heisenberg nilmanifold, first studied by Wen Huang, Jianya Liu and Ke Wang. We advance the work of Huang, Liu, Wang, and their followers to a broad generality by removing the previously imposed restrictive symmetry condition.

Möbius Disjointness Conjecture for a skew product on a circle and the Heisenberg nilmanifold

TL;DR

The paper proves Sarnak's Möbius disjointness conjecture for a skew product on the circle coupled with the 3-dimensional Heisenberg nilmanifold, removing a previous symmetry condition. The authors establish polynomial-rate rigidity for irrational rotation parameters and, under smoothness assumptions, derive sub-polynomial measure complexity, enabling Möbius disjointness via PR rigidity and HWY-type criteria. The approach blends continued-fraction analysis, harmonic analysis on the Heisenberg group, and a careful decomposition into resonant and non-resonant parts to control double exponential sums arising from the skew product. These results extend Möbius disjointness to a broader class of distal skew products on non-abelian nilmanifolds, contributing substantial evidence toward Sarnak's conjecture in a non-commutative setting.

Abstract

We establish Sarnak's conjecture on Möbius disjointness for the dynamical system of a skew product on a circle and the three-dimensional Heisenberg nilmanifold, first studied by Wen Huang, Jianya Liu and Ke Wang. We advance the work of Huang, Liu, Wang, and their followers to a broad generality by removing the previously imposed restrictive symmetry condition.
Paper Structure (34 sections, 8 theorems, 161 equations)

This paper contains 34 sections, 8 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Rigidity and Measure complexity
  3. PR rigidity
  4. Measure complexity
  5. Some preparation
  6. $3$-dimensional Heisenberg nilmanifold
  7. A metric on $\mathbb{T}\times \Gamma \backslash G$
  8. Continued fractions and Convergents
  9. Initial steps for the proof of Theorem \ref{['intd']}
  10. Formula for the iteration of $S$
  11. Bounding the metric on $\mathbb{T}\times \Gamma \backslash G$
  12. Treatments for $H_n(t)$
  13. Proof of Theorem \ref{['intd']}
  14. Proof of Lemma \ref{['lem31']}
  15. Case (i): Evaluation of $\Sigma_{n,0}$
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $\mathbb{T}$ be the unit circle and $\Gamma\backslash G$ the $3$-dimensional Heisenberg nilmanifold. Let $\alpha\in [0, 1)$ and $S_\alpha$ be the skew product on $\mathbb{T}\times\Gamma\backslash G$ defined in S, where $\varphi, \eta, \psi$ are periodic functions from $\mathbb{R}$ to $\mathbb{R}

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 5.1
  • Lemma 10.1
  • proof