arXiv:2006.07646v2

Sarnak's Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow

el Houcein el Abdalaoui, Mahesh Nerurkar

math.DSmath.NT37A3054H2047A3511N64

Paper metadata

Original submission: June 13, 2020
Latest revision: July 30, 2026 · version 2
Claimed MSC: 37A30, 54H20, 47A35, 11N64
arXiv categories: math.DS, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 23 pages. We present an unpublished theorem of Veech which is closely related to our main result with complete proof. The notion of Quasi-factor plays a key role. In this version, we give more details on the proof of our main result. Comments and suggestions are welcome

Abstract

It is shown that Sarnak's Möbius orthogonality conjecture is fulfilled for the compact metric dynamical systems for which every invariant measure has singular spectra. This is accomplished by first establishing a special case of Chowla conjecture which gives a correlation between the Möbius function and its square. Then a computation of W. Veech, followed by an argument using the notion of `affinity between measures', (or the so-called `Hellinger method'), completes the proof. We further present an unpublished theorem of Veech which is closely related to our main result. This theorem asserts, if for any probability measure in the closure of the Cesaro averages of the Dirac measure on the shift of the Möbius function, the first projection is in the orthocomplement of its Pinsker algebra then Sarnak Möbius disjointness conjecture holds. Among other consequences, we obtain a simple proof of Matom{ä}ki-Radziwiłl-Tao's theorem and Matom{ä}ki-Radziwiłl's theorem on the correlations of order two of the Liouville function.

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Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements2 reported findingsContains unsupported statements

The main singular-spectrum disjointness theorem is not established because its decisive Rokhlin–Sinai sub-sigma-algebra is transferred from the Chowla–Sarnak–Veech measure to an arbitrary quasi-generic measure without a measure-class relation.

Theorem 3.1Not able to verify

Sarnak disjointness for all singular-spectrum systems is not established by Proposition 3.16

Pages 5 and 13–14 · Theorem 3.1 and Proposition 3.16 · arXiv:2006.07646v2

The proof requires a tail sigma-algebra and conditional-expectation identities inside the measure space of an arbitrary quasi-generic measure $\eta$ . The cited Rokhlin–Sinai construction supplies them only modulo the distinguished measure $\eta_M$ . Because $\eta$ may be singular to $\eta_M$ , the construction does not transfer, leaving the spectral exclusion on which Theorem 3.1 depends unproved.

Theorem 3.2 and Corollary 3.3Correct after the displayed typo correction

Möbius orthogonality to observables of the square-free shift

Pages 10–12 · Theorem 3.2 and Corollary 3.3 · arXiv:2006.07646v2

Interpreting the argument of $f$ as the shifted square-free configuration $S^n\mu^2$ , Theorem 3.2 follows from Besicovitch almost periodicity of $\mu^2$ and Davenport's uniform exponential-sum estimate for $\mu$ . Approximation of a continuous observable by cylinder functions reduces it to finitely many shifted square-free coordinates, each retaining Besicovitch approximation. Corollary 3.3 is the corresponding cylinder specialization. This independent part of the paper does not use the invalid measure-transfer step in Proposition 3.16.

02Proofs3 reported findingsContains incorrect or incomplete proofs

Proposition 3.16 uses a measure-dependent sigma-algebra and conditional expectations as though they were measure independent; this is a substantive invalid inference in the proof of the main theorem.

Proposition 3.16Incorrect as written

The Rokhlin–Sinai factor is used modulo the wrong measure

Pages 13–14 · proof of Proposition 3.16 · arXiv:2006.07646v2

The equality between a decreasing tail and the full sigma-algebra is established modulo $\eta_M$ . The proof then invokes the same equality and the associated conditional expectations in $L^2(\eta)$ , although no domination or equivalence between $\eta$ and $\eta_M$ is assumed. A set null for $\eta_M$ can have positive $\eta$ -measure, so this transfer is formally invalid. No verified repair is supplied.

Proof of Theorem 3.2Correct and complete after the notation correction

Besicovitch approximation and Davenport cancellation

Pages 10–12 · proof of Theorem 3.2 · arXiv:2006.07646v2

Rauzy's theorem supplies trigonometric-polynomial approximants to every finite product of shifts of $\mu^2$ in the $B^1$ seminorm. Multiplying such a polynomial by $\mu(n)e^{in\theta}$ reduces the average to finitely many exponential sums covered uniformly by Davenport's estimate. Cauchy–Schwarz controls the $B^1$ approximation error, and density of cylinder functions in the square-free subshift extends the conclusion to every continuous $f$ . The order of approximation and $N$ -limits is explicit and no Chowla-type input is assumed.

Theorem 3.2Typo · no status impact

A scalar is printed where an orbit point is required

Page 6 · statement of Theorem 3.2 · arXiv:2006.07646v2

The displayed expression applies $f$ to $\mu^2(n)$ , although $f$ is a function on the square-free subshift. The proof and surrounding formulas use $f$ evaluated at the shifted point $S^n\mu^2$ . Replacing the scalar by that orbit point is uniquely determined and does not alter the theorem.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2006.07646v2
Authors listed: el Houcein el Abdalaoui, Mahesh Nerurkar
Audit date: August 18, 2026

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Sarnak's Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Sarnak disjointness for all singular-spectrum systems is not established by Proposition 3.16

Möbius orthogonality to observables of the square-free shift

The Rokhlin–Sinai factor is used modulo the wrong measure

Besicovitch approximation and Davenport cancellation

A scalar is printed where an orbit point is required

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