Veech's conjecture on Möbius averages for bounded sequences with zero-entropy shift orbit closure

Veech's conjecture, in its bounded-sequence form, asserts that the Möbius function is orthogonal in Cesàro average to every bounded deterministic sequence: for every $f\in\ell_\infty(\mathbb{N})$ whose shift-orbit closure (hull) has zero topological entropy, $\frac1N\sum_{n\le N}\mu(n)f(n)\to0$. El Abdalaoui and Lin (2026) attribute the two-sided $\ell_\infty(\mathbb{Z})$ version of this statement to Veech (American Journal of Mathematics 139 (2017), p. 1158), call the one-sided $\ell_\infty(\mathbb{N})$ form above its adaptation, and prove as their Theorem 4.5 that both are equivalent to Sarnak's Möbius disjointness conjecture for all zero-entropy compact metric systems, including non-invertible ones. The conjecture is open: all proved cases cover strictly smaller classes (rotations by Davenport, systems with sub-polynomial measure complexity by Huang–Wang–Ye, tame and weakly tame systems by El Abdalaoui–Nerurkar, and other structured families), and no counterexample is known. A naming hazard: the distinct measure-theoretic 'Veech's conjecture' on Furstenberg systems of the Möbius function, proved by Kanigowski–Kułaga-Przymus–Lemańczyk–de la Rue (2023), is an equivalence meta-statement and does not close this problem.

Background

The Möbius function $\mu$ is the arithmetic function on positive integers defined by $\mu(1)=1$, $\mu(n)=(-1)^k$ when $n$ is a product of $k$ distinct primes, and $\mu(n)=0$ when $n$ is divisible by a square. Its average behavior is a fine probe of the multiplicative structure of the integers: the prime number theorem is equivalent to $\frac1N\sum_{n\le N}\mu(n)\to0$. Already in 1937 Davenport proved the stronger quantitative estimate $\sup_{\alpha}\left|\frac1N\sum_{n\le N}\mu(n)e^{2\pi i n\alpha}\right|=O(N^{-c})$, showing that $\mu$ is orthogonal to every pure oscillation; this is the archetype of the results discussed here.

A topological dynamical system is a pair $(X,T)$ where $X$ is a compact metric space and $T:X\rightarrow X$ is continuous ($T$ need not be invertible). Its topological entropy $h_{top}(X,T)$ is the standard nonnegative invariant measuring the exponential growth rate of the number of distinguishable orbit segments of $T$; $h_{top}=0$ is the formal counterpart of 'deterministic' as opposed to 'chaotic'. A sequence $(a_n)$ is generated by $(X,T)$ if $a_n=f(T^{n}x)$ for some point $x\in X$ and continuous observable $f:X\to\mathbb{C}$.

Sarnak's Möbius disjointness conjecture (formulated around 2010) predicts that $\mu$ is asymptotically orthogonal to every sequence generated by a zero-entropy system:

$$\lim_{N\to\infty}\frac1N\sum_{n=1}^{N}\mu(n)\,f(T^{n}x)=0$$

for every compact metric $(X,T)$ with $h_{top}(X,T)=0$, every $x\in X$, and every continuous $f$. The conjecture vastly generalizes Davenport's theorem, which corresponds to circle rotations, and is intertwined with the Chowla conjecture, which predicts that all correlations $\frac1N\sum_{n\le N}\mu(n+h_1)\cdots\mu(n+h_k)$ at distinct integer shifts $h_i$ vanish; a failure of Sarnak's conjecture would refute the Chowla conjecture.

The conjecture also has an intrinsic sequence formulation that removes the auxiliary system. Let $\ell_\infty(\mathbb{N})$ be the Banach space of bounded complex sequences with the supremum norm, and let $S$ be the left shift, $(Sf)(n)=f(n+1)$. For $f\in\ell_\infty(\mathbb{N})$, the hull $X_f$ — the set of all pointwise limits of subsequences of $(S^{k}f)_{k\ge0}$, equivalently the weak-* closure of the shift orbit in $\ell_\infty=(\ell_1)^{*}$ — is a compact metrizable shift-invariant set, and $(X_f,S|_{X_f})$ is the dynamical system generated by $f$. Call $f$ deterministic when $h_{top}(X_f,S|_{X_f})=0$. The two notions of determinism coincide: every observable of a zero-entropy system has a zero-entropy hull, and conversely $f$ is recovered from its own hull by evaluation at coordinate 0, $f(n)=\mathrm{ev}_{0}(S^{n}f)$. Hence the assertion $\frac1N\sum_{n\le N}\mu(n)f(n)\to0$ for every deterministic $f\in\ell_\infty(\mathbb{N})$ is exactly Sarnak's conjecture in sequence language, not a weaker variant.

The two-sided sequence formulation — the same assertion with $\ell_\infty(\mathbb{Z})$, the bounded bilateral sequences, in place of $\ell_\infty(\mathbb{N})$ — is credited to Veech through his 2017 work on Möbius orthogonality for generalized Morse–Kakutani flows (and his 2016 lecture notes); El Abdalaoui and Lin cite it at p. 1158 of the 2017 paper and prove, as their Theorem 4.5, the equivalence of: (i) the $\ell_\infty(\mathbb{N})$ statement above, which they call the adaptation of Veech's conjecture to one-sided sequences; (ii) Sarnak's conjecture for all compact metric zero-entropy systems, invertible or not; and (iii) the analogous $\ell_\infty(\mathbb{Z})$ statement attributed to Veech. Despite intense activity since 2010, the conjecture is verified only for structured subclasses of zero-entropy systems: systems all of whose invariant measures have sub-polynomial measure complexity (Huang–Wang–Ye; measure complexity is the growth rate of the minimal number of $1/n$-balls needed to cover a set of full measure — this class includes every sequence whose hull is separable in the norm topology of $\ell_\infty$), tame and weakly tame systems (systems whose observable algebra contains no infinite independent binary tree, a strong structural condition) and Veech's class $\mathbb{K}(T)$ (El Abdalaoui–Nerurkar), rotations and nilsystems (a nilsystem being a rotation on a nilmanifold, a compact homogeneous space built from a filtration of a nilpotent group by repeated commutators; classical), automatic sequences (sequences whose $n$-th term is the output of a finite automaton reading the base-$k$ digits of $n$), substitutions (sequences obtained by iterating a symbol-rewriting rule), and Toeplitz systems (shift systems in which every position of the defining sequence is eventually occupied by a periodic pattern) (surveyed by Ferenczi–Kułaga-Przymus–Lemańczyk). No argument is known for the unrestricted zero-entropy class, and no counterexample is known. A terminology warning: in Kanigowski–Kułaga-Przymus–Lemańczyk–de la Rue (2023) the phrase Veech's conjecture denotes a different, measure-theoretic statement about the Furstenberg systems of $\mu$.

Problem Statement

Prove or refute Veech's conjecture in its bounded-sequence form. Let $\mu$ be the Möbius function, defined by $\mu(1)=1$, $\mu(n)=(-1)^k$ when $n$ is a product of $k$ distinct primes, and $\mu(n)=0$ when $n$ is divisible by a square. Let $S$ be the left shift on $\ell_\infty(\mathbb{N})$ defined by $(Sf)(n)=f(n+1)$, and for $f\in\ell_\infty(\mathbb{N})$ let $X_f$ denote the hull of $f$, i.e. the closure of the orbit $\{S^{k}f:k\ge0\}$ in the topology of pointwise convergence (equivalently the weak-* topology of $\ell_\infty=(\ell_1)^{*}$), a compact metrizable $S$-invariant set. The conjecture asserts:

$$\text{for every } f\in\ell_\infty(\mathbb{N}) \text{ with } h_{top}(X_f,S|_{X_f})=0,\quad \lim_{N\to\infty}\frac1N\sum_{n=1}^{N}\mu(n)f(n)=0.$$

By Theorem 4.5 of El Abdalaoui–Lin (arXiv:2607.15960), this statement — the one-sided adaptation of the two-sided $\ell_\infty(\mathbb{Z})$ conjecture that those authors attribute to Veech — is equivalent to Sarnak's Möbius disjointness conjecture for every compact metric topological dynamical system with zero topological entropy, invertible or not; an answer in either direction therefore settles Sarnak's conjecture, and a negative answer would also refute the Chowla conjecture. Answers restricted to proper subclasses — for example norm-separable hulls, tame or weakly tame systems, systems of sub-polynomial measure complexity, rotations, automatic sequences, or Toeplitz systems — are strictly easier derived problems already treated in the literature and do not answer this question.

References

1. El Houcein El Abdalaoui and Michael Lin (2026). Operator ergodic theorems with Möbius "weights". arXiv:2607.15960. https://arxiv.org/abs/2607.15960 (parent paper establishing the equivalence cited above)
2. William A. Veech (2017). Möbius orthogonality for generalized Morse-Kakutani flows. American Journal of Mathematics 139(5), 1157–1203. DOI 10.1353/ajm.2017.0031. https://doi.org/10.1353/ajm.2017.0031 (p. 1158, as cited in El Abdalaoui–Lin, 2026)
3. Adam Kanigowski, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Thierry de la Rue (2023). On arithmetic functions orthogonal to deterministic sequences. Advances in Mathematics 428, 109138. DOI 10.1016/j.aim.2023.109138. arXiv:2105.11737. https://arxiv.org/abs/2105.11737
4. El Houcein El Abdalaoui and Mahesh Nerurkar (2023). Weakly tame systems, their characterizations and applications. Monatshefte für Mathematik 201(3), 725–769. DOI 10.1007/s00605-023-01861-y. arXiv:2001.05947. https://arxiv.org/abs/2001.05947 (the arXiv preprint version carries the slightly different title 'Weakly tame systems, their characterizations and application')
5. Wen Huang, Zhiren Wang, Xiangdong Ye (2019). Measure complexity and Möbius disjointness. Advances in Mathematics 347, 827–858. DOI 10.1016/j.aim.2019.03.007. arXiv:1707.06345. https://arxiv.org/abs/1707.06345
6. S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk (2018). Sarnak's Conjecture: What's New. In: Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, Lecture Notes in Mathematics 2213, 163–235. DOI 10.1007/978-3-319-74908-2_11. arXiv:1710.04039. https://arxiv.org/abs/1710.04039
7. El Houcein El Abdalaoui (2017, revised September 2022). On Veech's proof of Sarnak's theorem on the Möbius flow. arXiv:1711.06326. https://arxiv.org/abs/1711.06326
8. H. Davenport (1937). On some infinite series involving arithmetical functions (II). The Quarterly Journal of Mathematics os-8(1), 313–320. DOI 10.1093/qmath/os-8.1.313. https://doi.org/10.1093/qmath/os-8.1.313
9. Terence Tao (2016). Equivalence of the logarithmically averaged Chowla and Sarnak conjectures. arXiv:1605.04628. https://arxiv.org/abs/1605.04628