In this paper, we reduce the logarithmic Sarnak conjecture to the $\{0,1\}$-symbolic systems with polynomial mean complexity. By showing that the logarithmic Sarnak conjecture holds for any topologically dynamical system with sublinear complexity, we provide a variant of the $1$-Fourier uniformity conjecture, where the frequencies are restricted to any subset of $[0,1]$ with packing dimension less than one.
We show that the images via $z\mapsto z^m$ of the continuous part of the spectral measures of the dynamical systems generated by the 0-1 sequences of the Thue-Morse type are pairwise mutually singular for different odd numbers $m\in\N$. Sarnak's conjecture on orthogonality with the Möbius function is shown to hold for such dynamical systems. The same conjecture is shown to hold for all systems induced by regular Toeplitz sequences. A non-regular Toeplitz sequence for which Sarnak's conjecture fails is constructed.
In this paper, we show that for any sequence ${\bf a}=(a_n)_{n\in \Z}\in \{1,\ldots,k\}^\mathbb{Z}$ and any $ε>0$, there exists a Toeplitz sequence ${\bf b}=(b_n)_{n\in \Z}\in \{1,\ldots,k\}^\mathbb{Z}$ such that the entropy $h({\bf b})\leq 2 h({\bf a})$ and $\lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^N|a_n-b_n|<ε$. As an application of this result, we reduce Sarnak Conjecture to Toeplitz systems, that is, if the Möbius function is disjoint from any Toeplitz sequence with zero entropy, then the Sarnak conjecture holds.
Although Sarnak's conjecture holds for compact group rotations (irrational rotations, odometers), it is not even known whether it holds for all Jewett-Krieger models of such rotations. In this paper we show that it does, as long as the model is at the same a topological extension. In particular, we reestablish (after [AKL]) that regular Toeplitz systems satisfy Sarnak's conjecture, and, as another consequence, so do all generalized Sturmian subshifts (not only the classical Sturmian subshift). We also give an example of an irregular Toeplitz subshift which fits our criterion. We give an exampl…
An overview of last seven years results concerning Sarnak's conjecture on Möbius disjointness is presented, focusing on ergodic theory aspects of the conjecture.
We present Veech's proof of Sarnak's theorem on the Möbius flow which say that there is a unique admissible measure on the Möbius flow. As a consequence, we obtain that Sarnak's conjecture is equivalent to Chowla conjecture with the help of Tao's logarithmic Theorem which assert that the logarithmic Sarnak conjecture is equivalent to logaritmic Chowla conjecture, furthermore, if the even logarithmic Sarnak's conjecture is true then there is a subsequence with logarithmic density one along which Chowla conjecture holds, that is, the Möbius function is quasi-generic.
Motivated by the variations of Sarnak's conjecture due to El Abdalaoui, Kulaga-Przymus, Lemanczyk, De La Rue and by the observation that the Mobius function is a good weight (with limit zero) for the polynomial pointwise ergodic theorem in $L^q$, q>1, we introduce polynomial versions of the Sarnak conjecture for minimal systems.
We construct the counter-example for polynomial version of Sarnak's conjecture for minimal systems, which assets that the Möbius function is linearly disjoint from subsequences along polynomials of deterministic sequences realized in minimal systems. Our example is in the class of Toeplitz systems, which are minimal.
We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences of numbers in {-1,0,1}, and introduce several natural generalizations. We study the relationships between these properties and other notions from topological dynamics and ergodic theory.
Assuming the existence of Siegel zeros, we prove that there exists an increasing sequence of positive integers for which Chowla's Conjecture on $k$-point correlations of the Liouville function holds. This extends work of Germán and Kátai, where they studied the case $k=2$ under identical hypotheses. An immediate corollary, which follows from a well-known argument due to Sarnak, is that Sarnak's Conjecture on Möbius disjointness holds. More precisely, assuming the existence of Siegel zeros, there exists a subsequence of the natural numbers for which the Liouville function is asymptotically orth…
We show that the Möbius disjointess of zero entropy dynamical systems implies the existence of an increasing sequence of positive integers along which the Chowla conjecture on autocorrelations of the Möbius function holds.
We present in this paper a new method to deal with automatic sequences. This method allows us to prove a Möbius-randomness-principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences. Furthermore, we can show a Prime Number Theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to $0$.
Using techniques developed in \cite{KLR}, we verify Sarnak's conjecture for two classes of rank-one subshifts with unbounded cutting parameters. The first class of rank-one subshifts we consider are called {\em almost complete congruency classes} ({\em accc}), the definition of which is motivated by the main result of \cite{GS}, which implies that, when a rank-one subshift carries a unique non-atomic invariant probability measure, it is accc if it is measure-theoretically isomorphic to an odometer. The second class we consider consists of Katok's map and its generalizations.
In this paper, we give two precise definitions of a higher order oscillating sequence and show the importance of this concept in the study of Sarnak's conjecture. We prove that any higher order oscillating sequence of order $d$ is linearly disjoint from all affine distal flows on the $d$-torus for all $d\geq 2$. One consequence of this result is that any higher order oscillating sequence of order $2$ is linearly disjoint from all affine flows on the $2$-torus with zero topological entropy. In particular, this reconfirms Sarnak's conjecture for all affine flows on the $2$-torus with zero topolo…
The Möbius disjointness conjecture of Sarnak states that the Möbius 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 associ…
In this paper we study $C^*$-algebra version of Sarnak Conjecture for noncommutative toral automorphisms. Let $A_Θ$ be a noncommutative torus and $α_Θ$ be the noncommutative toral automorphism arising from a matrix $S\in GL(d,\mathbb{Z})$. We show that if the Voiculescu-Brown entropy of $α_Θ$ is zero, then the sequence $\{ρ(α_Θ^nu)\}_{n\in \mathbb{Z}}$ is a sum of a nilsequence and a zero-density-sequence, where $u\in A_Θ$ and $ρ$ is any state on $A_Θ$. Then by a result of Green and Tao, this sequence is linearly disjoint from the Möbius function.
We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many sign patterns which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the $κ-1$-Fourier …
We prove Veech's conjecture on the equivalence of Sarnak's conjecture on Möbius orthogonality with a Kolmogorov type property of Furstenberg systems of the M\''obius function. This yields a combinatorial condition on the Möbius function itself which is equivalent to Sarnak's conjecture. As a matter of fact, our arguments remain valid in a larger context: we characterize all bounded arithmetic functions orthogonal to all topological systems whose all ergodic measures yield systems from a fixed characteristic class (zero entropy class is an example of such a characteristic class) with the charac…
Assuming Sarnak conjecture is true for any singular dynamical process, we prove that the spectral measure of the Möbius function is equivalent to Lebesgue measure. Conversely, under Elliott conjecture, we establish that the Möbius function is orthogonal to any uniquely ergodic dynamical system with singular spectrum. Furthermore, using Mirsky Theorem, we find a new simple proof of Cellarosi-Sinai Theorem on the orthogonality of the square of the Möbius function with respect to any weakly mixing dynamical system. Finally, we establish Sarnak conjecture for a particular random model.
It is shown that there is an oscillating sequence of higher order which is not orthogonal to the class of dynamical flow with topological entropy zero. We further establish that any oscillating sequence of order $d$ is orthogonal to any $d$-nilsequence arising from the skew product on the $d$-dimensional torus $\mathbb{T}^d$. The proof yields that any oscillating sequence of higher order is orthogonal to any dynamical sequence arising from topological dynamical systems with quasi-discrete spectrum. however, we provide an example of oscillating sequence of higher order with large Gowers norms. …
Laboratoire de Mathématiques Raphaël Salem, UMR 5085 CNRS, Université de Rouen-Normandie（鲁昂诺曼底大学拉斐尔·萨勒姆数学实验室）；Department of Mathematics, Ben-Gurion University of the Negev（内盖夫本-古里安大学数学系）
We formulate the generalized Sarnak's Möbius disjointness conjecture for an arbitrary number field $K$, and prove a quantitative disjointness result between polynomial nilsequences $(Φ(g(n)Γ))_{n\in\mathbb{Z}^{D}}$ and aperiodic multiplicative functions on $\mathcal{O}_{K}$, the ring of integers of $K$. Here $D=[K\colon\mathbb{Q}]$, $X=G/Γ$ is a nilmanifold, $g\colon\mathbb{Z}^{D}\to G$ is a polynomial sequence, and $Φ\colon X\to \mathbb{C}$ is a Lipschitz function. The proof uses tools from multi-dimensional higher order Fourier analysis, multi-linear analysis, orbit properties on nilmanifold…
We show that Sarnak's conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.
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. Moreover, it is proved that the following classes of topological dynamical systems $(X,T)$ meet this condition and hence satisfy Sarnak's conjecture: (1) Each invariant Borel probability measure of $T$ has discrete spectrum. (2) $T$ is a homotopically trivial $C^\infty$ skew product system on $\mathbb{T}^2$ over an irrational ro…
In \cite{B} 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.
We prove a conjecture of Rudnick and Sarnak on the mass equidistribution of Hecke eigenforms. This builds upon independent work of the authors see arxiv.org:math/0809.1640 and arxiv.org:math/0809.1635.
Sarnak's Möbius disjointness conjecture states that Möbius function is disjoint to any zero entropy dynamics. We prove that Möbius disjointness conjecture holds for one-frequency analytic quasi-periodic cocycles which are almost reducible, which extend \cite{LS15,W17} to the noncommutative case. The proof relies on quantitative version of almost reducibility.
We show that Sarnak's conjecture on Möbius disjointness holds for all subshifts given by bijective substitutions and some other similar dynamical systems, e.g.\ those generated by Rudin-Shapiro type sequences.
Sarnak's Möbius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$, $\frac{1}{N} \sum_{n=1} ^N f(T^n x) μ(n)= o(1)$ 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 $μ(n)$ one can put any bounded sequence such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero.
We prove that Sarnak's conjecture holds for any infinite measure symbolic rank-one map. We further extended Bourgain-Sarnak's result, which says that the Möbius function is a good weight for the ergodic theorem, to maps acting on $σ$-finite measure spaces. We also discuss and extend Bourgain's theorem by establishing that there is a class of maps for which the Möbius disjointness property holds for any continuous bounded function. Our proof allows us to obtain an extension of Bourgain's theorem on Möbius disjointness for bounded rank one maps and a simple and self-contained proof of this fact.
A zero-one law for the spectral gap property of pairs in SU(2)
Mathematics Institute and DIMAP, University of Warwick（华威大学数学研究所和DIMAP）；Graduate School of Mathematical Sciences, the University of Tokyo（东京大学大学院数理科学研究科）
We show that Sarnak's conjecture on Möbius disjointness holds in every uniquely ergodic modelof a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $P\in\R[x]$ with irrational leading coefficient and for each multiplicative function $\bnu:\N\to\C$, $|\bnu|\leq1$, we have\[ \frac{1}{M} \sum\_{M\le m\textless{}2M} \frac{1}{H} \left| \sum\_{m\le n \textless{} m+H} e^{2πiP(n)}\bnu(n) \right|\longrightarrow 0 \] as $M\to\infty$, $H\to\infty$, $H/M\to 0$.
