Disjointness Conjecture Research Articles

Arbitrarily slow decay in the logarithmically averaged Sarnak conjecture

In 2017 Tao proposed a variant Sarnak's Möbius disjointness conjecture with logarithmic averaging: For any zero entropy dynamical system (X,T), 1log⁡N∑n=1Nf(Tnx)μ(n)n=o(1) for every f∈C(X) and every x∈X. We construct examples showing that this o(1) can go to zero arbitrarily slowly. Nonetheless, all of our examples satisfy the conjecture.

Möbius Disjointness Conjecture on the product of a circle and the Heisenberg nilmanifold

Let T be the unit circle and Γ﹨G the 3-dimensional Heisenberg nilmanifold. We prove that the Möbius function is linearly disjoint from a class of distal skew products on T×Γ﹨G. These results generalize a recent work of Huang-Liu-Wang.

Sarnak's Conjecture for nilsequences on arbitrary number fields and applications

We formulate the generalized Sarnak's Möbius disjointness conjecture for an arbitrary number field K, and prove a quantitative disjointness result between polynomial nilsequences (Φ(g(n)Γ))n∈ZD and aperiodic multiplicative functions on OK, the ring of integers of K. Here D=[K:Q], X=G/Γ is a nilmanifold, g:ZD→G is a polynomial sequence, and Φ:X→C is a Lipschitz function. This result, being a generalization of a previous theorem of the author in [44], requires a significantly different approach, which involves with multi-dimensional higher order Fourier analysis, multi-linear analysis, orbit properties on nilmanifold, and an orthogonality criterion of Kátai in OK.We also use variations of this result to derive applications in number theory and combinatorics: (1) we prove a structure theorem for multiplicative functions on K, saying that every bounded multiplicative function can be decomposed into the sum of an almost periodic function (the structural part) and a function with small Gowers uniformity norm of any degree (the uniform part); (2) we give a necessary and sufficient condition for the Gowers norms of a bounded multiplicative function in OK to be zero; (3) we provide partition regularity results over K for a large class of homogeneous equations in three variables. For example, for a,b∈Z﹨{0}, we show that for every partition of OK into finitely many cells, where K=Q(a,b,a+b), there exist distinct and non-zero x,y belonging to the same cell and z∈OK such that ax2+by2=z2.

Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions

We begin by establishing two ergodic theorems which have among their corollaries numerous classical results from multiplicative number theory, including the prime number theorem, a theorem of Pillai and Selberg, a theorem of Erdős and Delange, the mean value theorem of Wirsing, and special cases of the mean value theorem of Halász. Then, by building on the ideas behind our ergodic results, we recast Sarnak’s Möbius disjointness conjecture in a new dynamical framework. This naturally leads to an extension of Sarnak’s conjecture that focuses on the disjointness of actions of (N,+) and (N,⋅). We substantiate this extension by providing proofs of several special cases.

Arbitrarily slow decay in the Möbius disjointness conjecture

AbstractSarnak’s Möbius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$ , $({1}/{N})\! \sum _{n=1}^{N}\! f(T^{n} x) \mu (n)= o(1)$ for every $f\in \mathcal {C}(X)$ and every $x\in X$ . We construct examples showing that this $o(1)$ can go to zero arbitrarily slowly. In fact, our methods yield a more general result, where in lieu of $\mu (n)$ , one can put any bounded sequence $a_{n}$ such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero. Moreover, in our construction, the choice of x depends on the sequence $a_{n}$ but $(X,T)$ does not.

Disjointness of Möbius from asymptotically periodic functions

We investigate Sarnak's M\"obius Disjointness Conjecture through asymptotically periodic functions. It is shown that Sarnak's conjecture for rigid dynamical systems is equivalent to the disjointness of M\"obius from asymptotically periodic functions. We give sufficient conditions and a partial answer to the later one. As an application, we show that Sarnak's conjecture holds for a class of rigid dynamical systems, which improves an earlier result of Kanigowski-Lema{\'{n}}czyk-Radziwi{\l}{\l}.

Mean equicontinuity, complexity and applications

We will review the recent development of the research related to mean equicontinuity, focusing on its characterizations, its relationship with discrete spectrum, topo-isomorphy, and bounded complexity. Particularly, the application of the complexity function in the mean metric to the Sarnak and the logarithmic Sarnak Mobius disjointness conjecture will be addressed.

Hausdorff dimension of a class of three-interval exchange maps

<p style='text-indent:20px;'>In [<xref ref-type="bibr" rid="b5">5</xref>] Bourgain proves that Sarnak's disjointness conjecture holds for a certain class of three-interval exchange maps. In the present paper we slightly improve the Diophantine condition of Bourgain and estimate the constants in the proof. We further show that the new parameter set has positive, but not full Hausdorff dimension. This, in particular, implies that the Lebesgue measure of this set is zero.

Sarnak’s Conjecture for Irregular Flows on Infinite-dimensional Torus

Sarnak’s Disjointness Conjecture states that the Mobius function is disjoint with any zero-entropy flow. This note establishes this conjecture, with a rate, for Furstenberg’s irregular flows on the infinite-dimensional torus.

Measure complexity and Möbius disjointness

In this paper, the notion of measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's Möbius disjointness conjecture holds for any system for which every invariant Borel probability measure has sub-polynomial measure complexity. We then apply this result to a number of situations, including certain systems whose invariant measure don't all have discrete spectrum.

Furstenberg Systems of Bounded Multiplicative Functions and Applications

Abstract We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these systems have no irrational spectrum and their building blocks are Bernoulli systems and infinite-step nilsystems. One consequence of our structural result is that strongly aperiodic multiplicative functions satisfy the logarithmically averaged variant of the disjointness conjecture of Sarnak for a wide class of zero entropy topological dynamical systems, which includes all uniquely ergodic ones. We deduce that aperiodic multiplicative functions with values plus or minus one have super-linear block growth. Another consequence of our structural result is that products of shifts of arbitrary multiplicative functions with values on the unit disc do not correlate with any totally ergodic deterministic sequence of zero mean. Our methodology is based primarily on techniques developed in a previous article of the authors where analogous results were proved for the Möbius and the Liouville function. A new ingredient needed is a result obtained recently by Tao and Teräväinen related to the odd order cases of the Chowla conjecture.

Möbius disjointness for topological models of ergodic systems with discrete spectrum

We provide a criterion for a point satisfying the required disjointness condition in Sarnak's Möbius Disjointness Conjecture. As a direct application, we have that the conjecture holds for any topological model of an ergodic system with discrete spectrum.

Möbius disjointness conjecture for local dendrite maps

We prove that the Möbius disjointness conjecture holds for graph maps with zero topological entropy and for all monotone local dendrite maps. We further show that this also holds for continuous maps on certain class of dendrites. Moreover, we see that there is an example of transitive dendrite map with zero entropy for which Möbius disjointness conjecture holds.

Möbius disjointness for homogeneous dynamics

We prove Sarnak's M\"obius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. Namely, we show that if $G$ is any such group, $\Gamma\subset G$ a lattice, and $u\in G$ an Ad-unipotent element, then for every $x\in\Gamma\backslash G$ and every continuous, bounded function $f$ on $\Gamma\backslash G$, the sequence $f(xu^{n})$ cannot correlate with the M\"obius function on average.

Möbius disjointness for topological models of ergodic measure-preserving systems with quasi-discrete spectrum

By using a decomposition result for ergodic measure-preserving system with quasi-discrete spectrum, we prove that a generic point of an ergodic quasi-discrete spectrum measure in a topological dynamical system satisfies the required disjointness condition in Sarnak's Möbius Disjointness Conjecture. As a direct application, we have that Sarnak's Möbius Disjointness Conjecture holds for any topological model of an ergodic measure-preserving system with quasi-discrete spectrum.

Periods and factors of weak model sets

There is a renewed interest in weak model sets due to their connection to $\mathcal B$-free systems, which emerged from Sarnak's program on the M\"obius disjointness conjecture. Here we continue our recent investigation [arXiv:1511.06137] of the extended hull ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, a dynamical system naturally associated to a weak model set in an abelian group $G$ with relatively compact window $W$. For windows having a nowhere dense boundary (this includes compact windows), we identify the maximal equicontinuous factor of ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ and give a sufficient condition when ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ is an almost 1:1 extension of its maximal equicontinuous factor. If the window is measurable with positive Haar measure and is almost compact, then the system ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ equipped with its Mirsky measure is isomorphic to its Kronecker factor. For general nontrivial ergodic probability measures on ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, we provide a kind of lower bound for the Kronecker factor. All relevant factor systems are natural $G$-actions on quotient subgroups of the torus underlying the weak model set. These are obtained by factoring out suitable window periods. Our results are specialised to the usual hull of the weak model set, and they are also interpreted for ${\mathcal B}$-free systems.

The logarithmic Sarnak conjecture for ergodic weights

The Mobius disjointness conjecture of Sarnak states that the Mobius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness argument and the key ingredient is a structural result for measure preserving systems naturally associated with the Mobius and the Liouville function. We prove that such systems have no irrational spectrum and their building blocks are infinite-step nilsystems and Bernoulli systems. To establish this structural result we make a connection with a problem of purely ergodic nature via some identities recently obtained by Tao. In addition to an ergodic structural result of Host and Kra, our analysis is guided by the notion of strong stationarity which was introduced by Furstenberg and Katznelson in the early 90's and naturally plays a central role in the structural analysis of measure preserving systems associated with multiplicative functions.

On dynamics of graph maps with zero topological entropy

We explore the dynamics of graph maps with zero topological entropy. It is shown that a continuous map f on a topological graph G has zero topological entropy if and only if it is locally mean equicontinuous, that is the dynamics on each orbit closure is mean equicontinuous. As an application, we show that Sarnak’s Möbius disjointness conjecture is true for graph maps with zero topological entropy. We also extend several results known in interval dynamics to graph maps. We show that a graph map has zero topological entropy if and only if there is no 3-scrambled tuple if and only if the proximal relation is an equivalence relation; a graph map has no scrambled pairs if and only if it is null if and only if it is tame.