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

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.

💡 Research Summary

The paper proves Sarnak’s Möbius disjointness conjecture for a class of dynamical systems obtained by taking a skew product on the circle together with the three‑dimensional Heisenberg nilmanifold. The system was first studied by Huang, Liu and Wang; they required a restrictive symmetry condition φ = η on the two “vertical” functions appearing in the skew product. The present work removes this symmetry entirely and relaxes the smoothness assumptions to φ, η ∈ C^{2+2ε} and ψ ∈ C^{1+ε}, thereby covering a much broader family of skew products.

The authors begin by recalling Sarnak’s conjecture: for any topological dynamical system (X,T) with zero topological entropy, the Möbius function μ(n) is linearly disjoint from every observable f∘Tⁿ, i.e. the Cesàro average of μ(n)f(Tⁿx) tends to zero for all continuous f and all points x. Two modern tools for establishing this are (i) polynomial‑rate rigidity (PR‑rigidity) and (ii) sub‑polynomial measure complexity. PR‑rigidity means that for a dense family of observables there exists a sequence of times rₙ such that the L²‑norm of f∘T^{j rₙ}−f decays faster than any power of rₙ uniformly for |j|≤rₙ^{δ}. Sub‑polynomial measure complexity requires that the number of ε‑balls needed to cover almost all of X with respect to the averaged metric \bar dₙ grows slower than any polynomial in n.

The Heisenberg group G is realized as upper‑triangular 3×3 matrices with ones on the diagonal; its centre Z(G) consists of matrices with only the (1,3) entry non‑zero. A left‑invariant metric d_G is introduced via the ℓ∞‑norm on the coordinate map κ(g)=(x,y,z−xy). This metric descends to a bi‑invariant metric d_{Γ\G} on the nilmanifold Γ\G, where Γ is the integer lattice. The product space X=T×Γ\G is equipped with the Euclidean sum of the circle metric ∥·∥ and d_{Γ\G}.

The skew product S_α is defined by S_α(t, \bar g) = (t+α, \bar g·M(t)), where M(t) is a Heisenberg matrix whose entries involve the periodic functions φ(t), η(t), ψ(t). The n‑fold iterate S_αⁿ can be written explicitly using cumulative sums Φ_n(t)=∑{r=0}^{n-1}φ(t+rα), Ξ_n(t)=∑{r=0}^{n-1}η(t+rα), and a more complicated term Ψ_n(t) that mixes ψ with products of φ and η. These expressions control the displacement of the point in the nilmanifold after n steps.

The core of the proof is to exhibit a sequence {rₙ} such that the average squared distance ∫X d\bigl((t,g), S_α^{j rₙ}(t,g)\bigr)² dν is ≤ ε rₙ^{−ε} uniformly for |j|≤rₙ^{δ}. To achieve this, the authors exploit Diophantine properties of the rotation number α via its continued‑fraction convergents q_k. Classical estimates (e.g. ½q{k+1}^{−1}<∥q_kα∥<q_{k+1}^{−1}) give precise control of how well rational approximations approach α. Using these, together with the smoothness of φ, η, ψ, they bound the growth of Φ_n, Ξ_n, and Ψ_n and show that the non‑commutative central term of the Heisenberg group contributes only a negligible amount after averaging over the chosen rₙ. The C^{2+2ε} regularity of φ and η guarantees that their second derivatives are bounded, which in turn yields a decay of Fourier coefficients faster than any polynomial; the C^{1+ε} regularity of ψ suffices for the linear term.

Having established the decay (4.1), the definition of PR‑rigidity (2.1) is satisfied for a dense set of Lipschitz observables, and Proposition 2.1 then yields Möbius disjointness for every invariant measure ν. This proves Theorem 1.1 (PR‑rigidity) and, via Proposition 2.1, Theorem 2 (full Sarnak conjecture) for all α, both rational and irrational.

In addition, when φ, η, ψ are C^{∞}, the authors prove Theorem 1.3: the measure complexity s_n(X,T,d,ν,ε) grows sub‑polynomially for any invariant measure ν and irrational α. This is done by refining the estimates on the cumulative sums and exploiting the rapid decay of all derivatives of the functions involved. Proposition 2.2 then gives an alternative route to Möbius disjointness, confirming the conjecture under stronger smoothness but without any symmetry condition.

The paper concludes by comparing its results with prior work. Huang‑Liu‑Wang required φ=η and C^{∞} smoothness; subsequent papers (He‑Wang, Ma‑Wu) weakened smoothness but kept some form of symmetry (e.g., η=kφ). The present work eliminates the symmetry entirely and only needs modest smoothness, thereby establishing Sarnak’s conjecture for the most general skew product on T×Heisenberg nilmanifold considered to date. This advances the understanding of Möbius disjointness in non‑abelian, non‑commutative settings and opens the way for further extensions to higher‑step nilmanifolds and more general skew products.