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arxiv: 2604.16840 · v1 · submitted 2026-04-18 · 🧮 math.NT

M\"obius disjointness conjecture for Furstenberg's flow on mathbb{T}^ω in short intervals

Shuyang He , Qingyang Liu , Jing Ma This is my paper

Pith reviewed 2026-05-10 07:16 UTC · model grok-4.3

classification 🧮 math.NT
keywords Möbius disjointnessSarnak conjectureFurstenberg flowinfinite-dimensional torusshort intervalsDiophantine conditionsanalytic periodic functions
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The pith

The Möbius disjointness conjecture holds for Furstenberg's flow on the infinite-dimensional torus in short intervals of length at least N to the 5/8 power.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that Sarnak's Möbius disjointness conjecture applies to Furstenberg's flow on the infinite torus T^ω. The flow adds α to the first coordinate and then applies an analytic 1-periodic function h to shifted versions of the previous coordinate using an irrational β. The proof establishes that the sum of the Möbius function times any continuous observable along the orbit vanishes in short intervals (N-M, N] when M lies between N^{5/8+ε} and N. A sympathetic reader cares because the result covers an irregular system where Birkhoff averages need not exist and supplies the technically stronger short-interval version of orthogonality.

Core claim

For the flow T on T^ω given by successive coordinate additions T(x_1, x_2, …) = (x_1 + α, x_2 + h(x_1), x_3 + h(x_1 + β), …) with α satisfying Diophantine conditions, β irrational, and h analytic and 1-periodic, the average of μ(n) f(T^n x) over intervals of length M tends to zero whenever N^{5/8+ε} ≤ M ≤ N.

What carries the argument

Furstenberg's infinite-dimensional flow on T^ω, which encodes cumulative irrational shifts by multiples of β to produce irregular orbits while allowing analytic control over discrepancies.

If this is right

  • Continuous functions evaluated along the orbit remain orthogonal to the Möbius function when averaged over intervals longer than N to the 5/8 power.
  • The result applies to this irregular flow even though Birkhoff averages fail to exist at some points.
  • The short-interval orthogonality strengthens the classical full-interval version of Sarnak's conjecture for this system.
  • The proof technique yields uniform control for almost every starting point in T^ω.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar short-interval methods might adapt to other infinite-dimensional or non-uniquely ergodic systems constructed by successive coordinate perturbations.
  • The exponent 5/8 could be lowered if sharper discrepancy bounds for the flow become available.
  • The result links arithmetic orthogonality questions to the study of uniform distribution on infinite tori.

Load-bearing premise

The parameters α must obey specific Diophantine conditions, β must be irrational, and h must be analytic and 1-periodic so that orbit discrepancies remain controllable enough for short-interval estimates.

What would settle it

A concrete choice of α, β, h satisfying the stated conditions, together with a point x and continuous f, for which the normalized short-interval sum of μ(n) f(T^n x) fails to tend to zero in some sequence of intervals with length at least N^{5/8+ε}.

read the original abstract

Furstenberg's flow on the infinite-dimensional torus $\mathbb{T}^\omega$ is defined by \[ T (x_1, x_2, \ldots, x_\nu, \ldots) = (x_1 + \alpha, x_2 + h(x_1), \ldots, x_\nu + h(x_1 + (\nu-2)\beta), \ldots) \] with $\alpha\in \mathbb{R}$ satisfying certain diophantine conditions, $\beta\in \mathbb{R}\backslash\mathbb{Q},$ 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 $x\in \mathbb{T}^\omega$, and it is a generalization of Furstenberg's irregular flow on $\mathbb{T}^2$. The main result of this paper is that the M\"{o}bius Disjointness Conjecture of Sarnak holds for the above flow $(\mathbb{T}^\omega, T)$ in short intervals $(N-M, N]$ with $N^{5/8+\varepsilon} \leqslant M\leqslant N$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves that Sarnak's Möbius disjointness conjecture holds for the explicitly defined Furstenberg flow on the infinite-dimensional torus T^ω (with α satisfying Diophantine conditions, β irrational, and h 1-periodic analytic) in short intervals (N-M,N] for N^{5/8+ε} ≤ M ≤ N. The argument proceeds by establishing o(M) bounds on the relevant correlation sums via Fourier decay of h and orbit discrepancy estimates adapted to the infinite product structure.

Significance. If the result holds, it supplies a new, non-trivial instance of Möbius orthogonality for an irregular dynamical system of infinite topological dimension, extending the finite-dimensional Furstenberg example and the known short-interval results for regular systems. The short-interval range down to N^{5/8+ε} is a concrete quantitative advance that relies on the analyticity assumption to obtain sufficient decay.

minor comments (3)
  1. [Introduction] The precise Diophantine conditions imposed on α are only alluded to as 'certain' in the abstract and introduction; they should be stated explicitly (with the implied constants) already in §1 so that the reader can verify the Fourier-decay estimates without searching later sections.
  2. [§1] In the definition of the flow (equation (1.1)), the indexing of the coordinates begins with x_1 + α and x_2 + h(x_1), after which the shift by (ν-2)β appears; a short remark clarifying the convention for ν = 1,2 would prevent any ambiguity in the infinite product.
  3. [Proof of Theorem 1.2] The proof of the main theorem invokes an infinite-product measure on T^ω; a brief paragraph confirming that the correlation sums converge absolutely under the stated analyticity of h would make the passage from finite-dimensional approximations to the infinite case fully transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive recommendation for minor revision. The referee's summary accurately reflects the main result: an extension of Sarnak's Möbius disjointness conjecture to the explicitly constructed Furstenberg flow on the infinite torus, valid in short intervals down to length N^{5/8+ε}.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the flow explicitly via the given recurrence on T^ω with stated Diophantine conditions on α, irrationality of β, and analyticity of h. It then proves the short-interval Möbius orthogonality directly from these assumptions using Fourier decay and discrepancy estimates. No step reduces a claimed prediction to a fitted parameter by construction, no load-bearing premise rests solely on self-citation, and no ansatz or uniqueness theorem is smuggled in via prior author work. The derivation is self-contained against external benchmarks in ergodic theory and analytic number theory.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the Diophantine conditions on α, irrationality of β, and analyticity of h, which are stated as part of the flow definition rather than derived.

axioms (3)
  • domain assumption α satisfies certain Diophantine conditions
    Invoked in the definition of the flow to ensure the required approximation properties.
  • domain assumption β is irrational
    Required for the flow to be irregular as stated.
  • domain assumption h is 1-periodic and analytic
    Part of the flow definition to guarantee smoothness for the estimates.

pith-pipeline@v0.9.0 · 5523 in / 1479 out tokens · 52819 ms · 2026-05-10T07:16:26.124542+00:00 · methodology

discussion (0)

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Reference graph

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