Möbius disjointness conjecture for Furstenberg's flow on $\mathbb{T}^ω$ in short intervals

Abstract: Furstenberg's flow on the infinite-dimensional torus $\mathbb{T}<sup>ω$ is defined by [ T (x_1, x_2, \ldots, x_ν, \ldots) = (x_1 + α, x_2 + h(x_1), \ldots, x_ν+ h(x_1 + (ν-2)β), \ldots) ] with $α\in \mathbb{R}$ satisfying certain diophantine conditions, $β\in \mathbb{R}\backslash\mathbb{Q},$ and $h: \mathbb{R}\to \mathbb{R}$ being $1$-periodic and analytic. This flow is irregular in the sense that its Birkhoff average does not exist for some $x\in \mathbb{T}<sup>ω$, and it is a generalization of Furstenberg's irregular flow on $\mathbb{T}<sup>2$. The main result of this paper is that the Möbius Disjointness Conjecture of Sarnak holds for the above flow $(\mathbb{T}^ω, T)$ in short intervals $(N-M, N]$ with $$N^{{5/8+\varepsilon}} \leqslant M\leqslant N$$.

• The paper establishes Möbius disjointness for Furstenberg-type skew product flows on the infinite-dimensional torus over short intervals.
• It employs harmonic analysis, Fourier techniques, and Diophantine approximation to control irregular flows and achieve uniform estimates.
• The findings extend disjointness results to settings with minimal smoothness assumptions, reinforcing robustness in zero-entropy dynamics.

Möbius Disjointness for Furstenberg's Flow on $\mathbb{T}^\omega$ in Short Intervals

Introduction and Theoretical Context

The paper addresses Sarnak's Möbius Disjointness Conjecture (MDC) for a class of irregular skew product flows on the infinite-dimensional torus $\mathbb{T}^\omega$. This context extends a rich line of inquiry at the intersection of analytic number theory, dynamical systems, and ergodic theory. At the core of the discussion is the Möbius function $\mu(n)$ and its conjectured linear disjointness from any zero-entropy topological dynamical system, i.e., for any continuous $f$ and any $x$,

$$\lim_{N\to\infty} \frac1N \sum_{n\leq N} \mu(n) f(T^n x) = 0.$$

This conjecture, rooted in Sarnak's foundational work, is well-established for a variety of regular (uniquely ergodic) flows. However, the behavior of irregular flows—those for which certain pointwise (Birkhoff) averages fail to exist—remains a locus of intricate problems. The infinite-dimensional generalization under study draws from earlier investigations of skew products on $\mathbb{T}^2$ by Furstenberg and subsequent extensions by Liu, Sarnak, and others.

Main Results

The authors rigorously establish Möbius disjointness for a broad class of Furstenberg-type skew product flows on $\mathbb{T}^\omega$ in the highly nontrivial setting of short intervals. The central result is as follows: for the skew product flow

$$T(x_1, x_2, \ldots, x_\nu, \ldots) = (x_1+\alpha, x_2+h(x_1), x_3 + h(x_1 + \beta), \ldots, x_\nu + h(x_1 + (\nu-2)\beta), \ldots)$$

on $\mathbb{T}^\omega$, with $\mathbb{T}^\omega$0 irrational, and $\mathbb{T}^\omega$1 $\mathbb{T}^\omega$2-periodic and analytic, and for any short interval $\mathbb{T}^\omega$3 with $\mathbb{T}^\omega$4,

$\mathbb{T}^\omega$5

for any $\mathbb{T}^\omega$6 and any $\mathbb{T}^\omega$7. The result also extends—with quantitive rates—to cases with $\mathbb{T}^\omega$8 rational or irrational with Diophantine properties, and for $\mathbb{T}^\omega$9 of finite smoothness.

Strong uniformity in both the function $\mu(n)$0 and the basepoint $\mu(n)$1 is achieved. The technical assumptions on $\mu(n)$2 are notably relaxed compared to previous works, requiring only analyticity (or sufficient smoothness) rather than fast decay in Fourier coefficients.

Technical Methods

The proof combines harmonic analysis on infinite-dimensional compact groups with Fourier expansion techniques and subtle Diophantine approximation arguments. The authors leverage the decomposition of $\mu(n)$3 into resonant and non-resonant Fourier components, adapting a strategy previously effective for finite-dimensional tori and extending it to the infinite-dimensional setting.

A pivotal lemma (tracing to Wang and Yao) enables the non-resonant part of $\mu(n)$4 to be rewritten as a coboundary, facilitating a vanishing-multiplier argument via Fourier analysis and the mean-value zero property of the Möbius function. The authors deploy uniform estimates on exponential sums over primes in short intervals, specifically using strong bounds (after Zhan) to control sum magnitudes irrespective of the underlying frequency. The enumeration over the dual group of $\mu(n)$5—$\mu(n)$6—is central to the analysis, with significant combinatorial and analytic effort expended to ensure all frequencies (i.e., trigonometric monomials or characters) are controlled uniformly, yielding density in $\mu(n)$7.

Distality of the flow is established in full generality (without continuity of $\mu(n)$8), showing positive lower bounds on orbits' distance and thereby justifying application of Sarnak's framework.

Implications and Future Directions

The paper advances the frontier of Möbius disjointness into irregular, infinite-dimensional deterministic systems, showing that even in the absence of regularity (i.e., nonexistence of all Birkhoff averages), strong orthogonality to Möbius persists at the scale of short intervals. This highlights the robustness of arithmetic randomness in zero-entropy dynamics and provides a high-dimensional analog of earlier rigidity phenomena seen in finite-dimensional distal flows.

The extension to short intervals is technically significant, as it aligns with the best known ranges in the quantitative distribution of arithmetic functions—far beyond mean-square or logarithmic-averaged settings. The approach and outcomes further raise the prospect of treating more general models of skew products, possibly including nilflows or more elaborate distal systems, and with minimal smoothness hypotheses on the cocycle.

On the theoretical side, the techniques suggest that Möbius disjointness is a robust property in distal flows, even when complexity is augmented through infinite-dimensionality and irregular phase functions. Practically, these results reinforce the heuristic that in dynamical contexts with zero entropy, the Möbius function "behaves" like a random signal, largely immune to deterministic irregularities in the underlying flow.

From an analytic number theory perspective, the bounds established here for short-interval sums could inform further results on the correlations of arithmetic functions with highly irregular sequences or flows. The analytical framework—especially the marriage of Fourier analytic and Diophantine methods—should inspire new tools for approaching Sarnak-type conjectures in even broader classes of non-regular systems, including those arising naturally in homogeneous dynamics or symbolic models.

Conclusion

This paper achieves a comprehensive extension of the Möbius disjointness conjecture to the class of Furstenberg's irregular skew products on the infinite-dimensional torus, covering both analytic and finitely smooth cocycles and establishing strong quantitative results in short intervals. The technical innovations and generalizations suggest wide applicability and open new venues for both ergodic-theoretic and arithmetic investigations of randomness and determinism in dynamical systems.