Veech's conjecture for the Möbius flow: uniform orthogonality of the coordinate projection to the Pinsker σ-algebra of every quasi-generic measure

Let $\boldsymbol{\mu}$ be the Möbius function, viewed as a point of the compact admissible subshift $X_{\mathcal{A}_3}\subset\{-1,0,1\}^{\mathbb{Z}}$ under the shift $S$, and let $\mathcal{I}_S(\boldsymbol{\mu})$ be the set of $S$-invariant measures obtained as weak-* limits of the empirical measures of the Möbius orbit (its Furstenberg systems, or quasi-generic measures). Veech conjectured that the zeroth coordinate projection $\operatorname{pr}_1$ is orthogonal to the $L^2$-space of the Pinsker σ-algebra $\Pi_\eta(S)$ of every $\eta\in\mathcal{I}_S(\boldsymbol{\mu})$. Only the single Chowla–Sarnak–Veech (CSV) measure case is known. Kanigowski–Kulaga-Przymus–Lemańczyk–de la Rue (2021) proved that this uniform orthogonality condition is equivalent to Sarnak's Möbius disjointness conjecture, so the problem is exactly as hard as Sarnak's conjecture and remains open.

Background

The Möbius function is defined by $\mu(n)=0$ if $n$ is divisible by the square of a prime and $\mu(n)=(-1)^k$ if $n$ is a product of $k$ distinct primes; it is the central example of a deterministic-looking arithmetic function whose orbit structure encodes deep number theory. Sarnak's Möbius disjointness conjecture (formulated around 2010) asserts that for every homeomorphism $T$ of a compact metric space $X$ with zero topological entropy (topological entropy measures the exponential growth rate of orbit-distinguishable states; zero entropy is a strong form of determinism), and every continuous function $f\in C(X)$ and point $x\in X$,

$$\lim_{N\to\infty}\frac{1}{N}\sum_{n\le N}\mu(n)\,f(T^{n}x)=0,$$

that is, the Möbius sequence is orthogonal to every deterministic observable. The conjecture is open in general; it has been verified for many restricted classes of systems (for example systems with discrete spectrum, horocycle flows, and — by el Abdalaoui–Nerurkar (arXiv:2006.07646) — systems all of whose invariant measures have singular spectrum).

The ergodic-theoretic reformulation studies the Möbius flow: the orbit closure of the two-sided Möbius sequence under the left shift $S$ inside $\{-1,0,1\}^{\mathbb{Z}}$. Because $\mu(n)=0$ whenever $p^2\mid n$, this closure is contained in the admissible subshift $X_{\mathcal{A}_3}$, where a sequence is called admissible (a notion going back to Mirsky) if for every prime $p$ the set of its nonzero positions misses at least one residue class modulo $p^2$. Writing $s$ for the coordinatewise squaring map, $s(x)_n=x_n^2$, the image subshift $X_{\mathcal{A}_2}=s(X_{\mathcal{A}_3})\subset\{0,1\}^{\mathbb{Z}}$ carries the squarefree-support sequence $\boldsymbol{\mu}^2$, and Mirsky's theorem states that $\boldsymbol{\mu}^2$ is generic for a uniquely ergodic measure $\nu_M$ on $X_{\mathcal{A}_2}$, the Mirsky measure. The statistical limit behaviors of $\boldsymbol{\mu}$ itself are encoded in its Furstenberg systems (also called quasi-generic measures): the elements of $\mathcal{I}_S(\boldsymbol{\mu})$, the nonempty compact set of $S$-invariant probability measures on $X_{\mathcal{A}_3}$ arising as weak-* limits along subsequences of the empirical measures $\frac{1}{N}\sum_{n\le N}\delta_{S^{n}\boldsymbol{\mu}}$. A priori $\mathcal{I}_S(\boldsymbol{\mu})$ may contain many measures; under the Chowla conjecture (the assertion that all self-correlations $\frac{1}{N}\sum_{n\le N}\mu(n+h_1)\cdots\mu(n+h_k)$ of the Möbius function vanish), which is itself open, it collapses to a single measure.

For a measure-preserving system $(X,\mathcal{B},\eta,S)$, the Pinsker σ-algebra $\Pi_\eta(S)$ is the largest $S$-invariant sub-σ-algebra on which the Kolmogorov–Sinai entropy (the invariant measuring the rate of information production) vanishes; it isolates the deterministic part of the system, and its orthocomplement in $L^2$ carries all the randomness. The Sarnak–Veech theorem singles out one distinguished admissible measure on $X_{\mathcal{A}_3}$: there exists a unique ergodic admissible measure $\eta_M$ (the Chowla–Sarnak–Veech, or CSV, measure; Veech calls it the Chowla measure) whose Pinsker σ-algebra is exactly $\Pi(S)=s^{-1}(\mathcal{B}(X_{\mathcal{A}_2}))$ and for which the zeroth coordinate projection $\operatorname{pr}_1$ satisfies $\mathbb{E}(\operatorname{pr}_1\mid\Pi(S))=0$, i.e. $\operatorname{pr}_1\in L^2(X_{\mathcal{A}_3},\Pi(S),\eta_M)^{\perp}$; moreover every $\eta\in\mathcal{I}_S(\boldsymbol{\mu})$ satisfies $\Pi_\eta(S)\supset\Pi(S)$ and projects to $\nu_M$ under $s$. Whether $\eta_M$ itself belongs to $\mathcal{I}_S(\boldsymbol{\mu})$ is equivalent to the Chowla conjecture.

In unpublished 2016 lecture notes and a private communication (documented in el Abdalaoui's arXiv:1711.06326 and in arXiv:2006.07646), W. Veech proved that if $\operatorname{pr}_1$ is orthogonal to the $L^2$-space of the Pinsker σ-algebra of every $\eta\in\mathcal{I}_S(\boldsymbol{\mu})$, then Sarnak's conjecture holds, and he conjectured the converse implication. Kanigowski, Kulaga-Przymus, Lemańczyk, and de la Rue (arXiv:2105.11737) proved this equivalence: the uniform Pinsker-orthogonality over all Furstenberg systems of $\boldsymbol{\mu}$ holds if and only if Sarnak's Möbius disjointness conjecture holds, and Sarnak's conjecture would moreover force every Furstenberg system of $\boldsymbol{\mu}$ to have positive entropy. Terminology: in this task the name "Veech's conjecture" denotes the orthogonality condition itself, i.e. the requirement that $\operatorname{pr}_1\in L^2(X_{\mathcal{A}_3},\Pi_\eta(S),\eta)^{\perp}$ for every $\eta\in\mathcal{I}_S(\boldsymbol{\mu})$ (this is Conjecture 5.1 of arXiv:2006.07646 and, in the vocabulary of arXiv:2105.11737, the Veech condition with respect to the zero-entropy class); the equivalence of this condition with Sarnak's conjecture, whose proof is reported above, is a different statement and is not the object to be settled here. The strongest unconditional number-theoretic input available (Matomäki–Radziwiłł's short-interval estimates and the averaged two-point Chowla conjecture of Matomäki–Radziwiłł–Tao) settles only special system classes, not the orthogonality over the whole family $\mathcal{I}_S(\boldsymbol{\mu})$ at once.

Problem Statement

Prove or disprove Veech's conjecture for the Möbius flow. Let $\boldsymbol{\mu}$ denote the Möbius function ($\mu(n)=0$ if $n$ is divisible by the square of a prime, $\mu(n)=(-1)^k$ if $n$ is a product of $k$ distinct primes), viewed as a point of the compact subshift $X_{\mathcal{A}_3}\subset\{-1,0,1\}^{\mathbb{Z}}$ under the shift $S$; here $X_{\mathcal{A}_3}$ is the admissible subshift, meaning that for every prime $p$ the set of nonzero positions of any sequence in $X_{\mathcal{A}_3}$ misses at least one residue class modulo $p^2$. Let $\mathcal{I}_S(\boldsymbol{\mu})$ be the set of $S$-invariant probability measures obtained as weak-* limits along subsequences of the empirical measures $\frac{1}{N}\sum_{n\le N}\delta_{S^{n}\boldsymbol{\mu}}$ (the quasi-generic, or Furstenberg, measures of $\boldsymbol{\mu}$). The problem asks whether, for every $\eta\in\mathcal{I}_S(\boldsymbol{\mu})$, the coordinate projection $\operatorname{pr}_1$, characterized by $\operatorname{pr}_1(S^{n}\boldsymbol{\mu})=\boldsymbol{\mu}(n)$, belongs to the orthocomplement $L^2(X_{\mathcal{A}_3},\Pi_\eta(S),\eta)^{\perp}$, where $\Pi_\eta(S)$ is the Pinsker σ-algebra of the system $(X_{\mathcal{A}_3},\mathcal{B}(X_{\mathcal{A}_3}),\eta,S)$ — the largest $S$-invariant sub-σ-algebra on which the Kolmogorov–Sinai entropy vanishes; equivalently,

$$\mathbb{E}_\eta\big(\operatorname{pr}_1\,\big|\,\Pi_\eta(S)\big)=0\quad\text{for every }\eta\in\mathcal{I}_S(\boldsymbol{\mu}).$$

The quantification over all quasi-generic measures is essential: orthogonality is known only for the single Chowla–Sarnak–Veech measure $\eta_M$ (Sarnak–Veech theorem). By the theorem of Kanigowski–Kulaga-Przymus–Lemańczyk–de la Rue (Corollary 1.3 of arXiv:2105.11737), this statement is equivalent to Sarnak's Möbius disjointness conjecture, so any complete answer (proof or refutation) settles Sarnak's conjecture itself; a proof would additionally imply that every Furstenberg system of $\boldsymbol{\mu}$ has positive Kolmogorov–Sinai entropy.

References

1. el Houcein el Abdalaoui and Mahesh Nerurkar, 'Sarnak's Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow', 2020 (v2 posted 30 July 2026), arXiv:2006.07646, DOI 10.48550/arXiv.2006.07646, https://arxiv.org/abs/2006.07646
2. Adam Kanigowski, Joanna Kulaga-Przymus, Mariusz Lemańczyk, and Thierry de la Rue, 'On arithmetic functions orthogonal to deterministic sequences', 2021, arXiv:2105.11737, DOI 10.48550/arXiv.2105.11737, https://arxiv.org/abs/2105.11737
3. el Houcein el Abdalaoui, 'On Veech's proof of Sarnak's theorem on the Möbius flow', 2017 (v5 revised 15 September 2022), arXiv:1711.06326, DOI 10.48550/arXiv.1711.06326, https://arxiv.org/abs/1711.06326
4. El Houcein El Abdalaoui and Michael Lin, 'Operator ergodic theorems with Möbius "weights"', 2026, arXiv:2607.15960, DOI 10.48550/arXiv.2607.15960, https://arxiv.org/abs/2607.15960
5. el Houcein el Abdalaoui, 'On the spectral analysis of dynamical Möbius-Sarnak process and topological entropy of Möbius fonction' (title as posted on arXiv), 2026, arXiv:2607.29275, DOI 10.48550/arXiv.2607.29275, https://arxiv.org/abs/2607.29275
6. Kaisa Matomäki, Maksym Radziwiłł, and Terence Tao, 'An averaged form of Chowla's conjecture', Algebra & Number Theory 9 (2015), 2167–2196, DOI 10.2140/ant.2015.9.2167, https://arxiv.org/abs/1503.05121
7. Kaisa Matomäki and Maksym Radziwiłł, 'Multiplicative functions in short intervals', Annals of Mathematics (2) 183 (2016), 1015–1056, DOI 10.4007/annals.2016.183.3.6, https://arxiv.org/abs/1501.04585