Strong Möbius and Liouville MOMO property for the unipotent automorphism of the two-torus

The affine unipotent map $T(x,y)=(x,x+y)$ of the two-torus $\mathbb{T}^2$ is a zero-entropy algebraic dynamical system whose ordinary Möbius disjointness follows from classical estimates, yet it is not known whether it satisfies the strong MOMO property with respect to the Möbius function $\mu$ or the Liouville function $\lambda$: uniform cancellation over arbitrary finite unions of integer intervals whose lengths tend to infinity, with the supremum over the phase taken separately for each interval and the per-block suprema required to cancel on average. Kanigowski, Kułaga-Przymus, Lemańczyk and de la Rue, after proving Veech's conjecture on the equivalence of Sarnak-type orthogonality with a Kolmogorov property of Furstenberg systems for ec-classes, explicitly left this problem open (arXiv:2105.11737; Advances in Mathematics 2023). A sibling open question from the same paper — the extension of the Veech–Sarnak equivalence to arbitrary characteristic classes — has in the meantime been refuted by Górska–Lemańczyk–de la Rue (arXiv:2404.07907).

Background

The Möbius function $\mu(n)$ (equal to $0$ when $n$ is divisible by a square and to $(-1)^r$ when $n$ is a product of $r$ distinct primes) and the Liouville function $\lambda(n)=(-1)^r$, where $r$ counts the prime factors of $n$ with multiplicity, are the classical sign-valued multiplicative functions, and their cancellation properties measure the randomness of the primes. Sarnak's Möbius disjointness conjecture asserts that $\mu$ is orthogonal to every topological dynamical system $(X,T)$ of zero topological entropy (entropy measuring the exponential rate of orbit separation; zero entropy is the mildest complexity class), in the sense that $\frac{1}{N}\sum_{n<N}\mu(n)f(T^nx)\to 0$ for every continuous observable $f\in C(X)$ and every point $x\in X$. The prototype of such cancellation is the classical estimate of Davenport: $\sup_{\alpha\in\mathbb{T}}\big|\sum_{n<N}\mu(n)e^{2\pi i\alpha n}\big|\ll_A N/\log^A N$ for every $A>0$, where $\mathbb{T}=\mathbb{R}/\mathbb{Z}$; it yields Möbius disjointness for circle rotations.

For the Sarnak program one needs a strengthening introduced by Gomilko, Lemańczyk and de la Rue: the strong $u$-MOMO property. A topological system $(X,T)$ satisfies it (for a bounded arithmetic function $u:\mathbb{N}\to\mathbb{C}$) if for every $f\in C(X)$ and every increasing sequence of integers $(b_k)$ with $b_{k+1}-b_k\to\infty$,

$$\lim_{K\to\infty}\frac{1}{b_K}\sum_{k<K}\Big\|\sum_{b_k\le n<b_{k+1}}u(n)\,f\circ T^n\Big\|_{C(X)}=0,$$

where $\|\cdot\|_{C(X)}$ is the supremum over $x\in X$. The expression averages, over the blocks, the uniform norm of the orbital sum of $u$ over each single block; the cancellation is required uniformly in the choice of the pieces. Gomilko–Lemańczyk–de la Rue proved that Sarnak's conjecture is equivalent to all zero-entropy systems satisfying the strong $\mu$-MOMO property. The simplest system for which the property itself is open is the unipotent affine automorphism of the two-torus,

$T(x,y)=(x,x+y),$

a zero-entropy, distal (informally: distinct orbits never asymptotically approach each other) algebraic system. Its ordinary Möbius orthogonality is easy and even uniform in the starting point by Davenport's estimate, but applying the strong MOMO definition to $f(x,y)=e^{2\pi iy}$, for which $f(T^n(x,y))=e^{2\pi i(y+nx)}$, reduces the property to a purely arithmetic question: for every $(b_k)$ with $b_{k+1}-b_k\to\infty$,

$$\lim_{K\to\infty}\frac{1}{b_K}\sum_{k<K}\sup_{\alpha\in\mathbb{T}}\Big|\sum_{b_k\le n<b_{k+1}}u(n)e^{2\pi i\alpha n}\Big|=0.$$

Each block contributes its own phase supremum, so a Davenport-type bound must hold for every interval separately, uniformly in the phase and in the position and length of the interval, and the per-block suprema must still cancel on average. Known estimates either concern one interval with savings depending on its endpoint, or hold only on average over positions.

The problem also has a structural dynamical meaning. A Furstenberg system of $u$ is any weak-star limit point of the empirical measures $\frac{1}{N_k}\sum_{n<N_k}\delta_{S^nu}$, where $S$ is the shift on the space of one-sided sequences; such systems encode the correlation structure of $u$. Kanigowski, Kułaga-Przymus, Lemańczyk and de la Rue proved that no measure-preserving system possessing a strongly $u$-MOMO topological model can occur as a Furstenberg system of $u$; hence a positive answer for $u=\lambda$ would settle a 2016 question of Frantzikinakis (recorded as Problem 3.1 of a 2016 CIRM workshop, and reproduced in that paper): show that the unipotent system is not a Furstenberg system of the Liouville function. For $u$ generic (Furstenberg systems existing along the full sequence of scales), the block condition above is equivalent, by results of Frantzikinakis reproduced in that paper together with an appendix of Kanigowski–Lemańczyk–Ulcigrai, to the vanishing of the order-2 uniformity seminorm $\|u\|_{u^2}$, the arithmetic-function analogue of the Gowers uniformity norms. The same paper proved Veech's conjecture — orthogonality of $u$ to all topological systems whose invariant measures yield members of a fixed characteristic class (a class of measure-preserving systems closed under isomorphisms and factors) is equivalent to a Kolmogorov-type (Veech) condition on the Furstenberg systems of $u$ — for classes closed under taking ergodic components, so-called ec-classes, and explicitly posed the strong MOMO problem for the unipotent system as open. Later work of Górska–Lemańczyk–de la Rue shows that the ec-assumption cannot be dropped in the Veech–Sarnak equivalence, so the surviving open core of the paper's questions is precisely the uniform cancellation problem for $\mu$ and $\lambda$ stated above.

Problem Statement

Let $u=\mu$ (the Möbius function) and $u=\lambda$ (the Liouville function). Determine whether the following assertion holds for each of the two functions: for every increasing sequence of integers $(b_k)_{k\ge1}$ with $b_{k+1}-b_k\to\infty$,

$$\lim_{K\to\infty}\frac{1}{b_K}\sum_{k<K}\sup_{\alpha\in\mathbb{T}}\Big|\sum_{b_k\le n<b_{k+1}}u(n)\,e^{2\pi i\alpha n}\Big|=0,$$

where $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Equivalently: does the unipotent automorphism $T(x,y)=(x,x+y)$ of $\mathbb{T}^2$ satisfy the strong $u$-MOMO property, that is, does

$$\lim_{K\to\infty}\frac{1}{b_K}\sum_{k<K}\Big\|\sum_{b_k\le n<b_{k+1}}u(n)\,f\circ T^n\Big\|_{C(\mathbb{T}^2)}=0$$

hold for every $f\in C(\mathbb{T}^2)$ and every such $(b_k)$? The two formulations are equivalent: taking $f(x,y)=e^{2\pi iy}$, for which $f(T^n(x,y))=e^{2\pi i(y+nx)}$, makes each block norm equal to the corresponding phase supremum in the first display, and the general case follows by expanding arbitrary observables in trigonometric polynomials and using uniform approximation. A positive answer for $u=\lambda$ would in particular prove that the unipotent system is not a Furstenberg system of the Liouville function (a 2016 question of Frantzikinakis recorded in the source paper), and for $u$ generic the block condition is equivalent to $\|u\|_{u^2}=0$. A complete answer determines, for each of the two functions separately, whether the displayed limit holds — either by proving it, or by disproving it, for instance by exhibiting a block sequence $(b_k)$ with $b_{k+1}-b_k\to\infty$ for which the displayed limsup is strictly positive.

References

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