Sarnak's Conjecture Research Articles

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.

Automatic sequences fulfill the Sarnak conjecture

We present in this paper a new method to deal with automatic sequences. This method allows us to prove a M\"obius-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 $0$.

Fully oscillating sequences and weighted multiple ergodic limit

We prove that fully oscillating sequences are orthogonal to multiple ergodic realizations of affine maps of zero entropy on compact Abelian groups. It is more than what Sarnak's conjecture requires for these dynamical systems.

Sequences from zero entropy noncommutative toral automorphisms and Sarnak Conjecture

In this paper we study C⁎-algebra version of Sarnak Conjecture for noncommutative toral automorphisms. Let AΘ be a noncommutative torus and αΘ be the noncommutative toral automorphism arising from a matrix S∈GL(d,Z). We show that if the Voiculescu–Brown entropy of αΘ is zero, then the sequence {ρ(αΘnu)}n∈Z is a sum of a nilsequence and a zero-density-sequence, where u∈AΘ and ρ is any state on AΘ. Then by a result of Green and Tao [9], this sequence is linearly disjoint from the Möbius function.

Erratum to “Automorphisms with Quasi-discrete Spectrum, Multiplicative Functions and Average Orthogonality Along Short Intervals”

We show that Sarnak's conjecture on M\"obius disjointness holds in every uniquely ergodic model of 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 \[ \frac{1}{M} \sum_{M\le m<2M} \frac{1}{H} \left| \sum_{m\le n < m+H} e^{2\pi iP(n)}\bnu(n) \right|\longrightarrow 0 \] as $M\to\infty$, $H\to\infty$, $H/M\to 0$.

Automorphisms with Quasi-discrete Spectrum, Multiplicative Functions and Average Orthogonality Along Short Intervals

We show that Sarnak's conjecture on M\obius disjointness holds in every uniquely ergodic model of 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 \[ \frac{1}{M} \sum_{M\le m<2M} \frac{1}{H} \left| \sum_{m\le n < m+H} e^{2\pi iP(n)}\bnu(n) \right|\longrightarrow 0 \] as $M\to\infty$, $H\to\infty$, $H/M\to 0$.

Automatic sequences generated by synchronizing automata fulfill the Sarnak conjecture

Automatic sequences generated by synchronizing automata fulfill the Sarnak conjecture

0-1 sequences of the Thue-Morse type and Sarnak’s conjecture

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 Mobius 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.

A polynomial version of Sarnak's conjecture

Motivated by the variations of Sarnak's conjecture due to El Abdalaoui, Kulaga-Przymus, Lemańczyk, de la Rue and by the observation that the Möbius function is a good weight (with limit zero) for the polynomial pointwise ergodic theorem, we introduce a polynomial version of the Sarnak conjecture for minimal systems.

Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture

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 sa

Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow

By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Möller and the present author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families:$$\begin{eqnarray}Z(s)=\det (1-{\mathcal{L}}_{s}^{+})\det (1-{\mathcal{L}}_{s}^{-}).\end{eqnarray}$$In this article we show that the operator families${\mathcal{L}}_{s}^{\pm }$arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for$s\in \mathbb{C}$,$\text{Re}s={\textstyle \frac{1}{2}}$, the operator${\mathcal{L}}_{s}^{+}$(respectively${\mathcal{L}}_{s}^{-}$) has a 1-eigenfunction if and only if there exists an even (respectively odd) Maass cusp form with eigenvalue$s(1-s)$. For non-arithmetic Hecke triangle groups, this result provides a new formulation of the Phillips–Sarnak conjecture on non-existence of even Maass cusp forms.

Endoscopy and cohomology growth on

AbstractWe apply the endoscopic classification of automorphic forms on $U(3)$ to study the growth of the first Betti number of congruence covers of a Picard modular surface. As a consequence, we establish a case of a conjecture of Sarnak and Xue on cohomology growth.

Theta Lifting and Cohomology Growth in p-adic Towers

We use the theta lift to study the multiplicity with which certain automorphic representations of cohomological type occur in a family of congruence covers of an arithmetic manifold. When the family of covers is a so-called p-adic congruence tower, we obtain sharp asymptotics for the number of representations that occur as lifts. When combined with theorems on the surjectivity of the theta lift due to Howe and Li, and Bergeron, Millson, and Moeglin, this allows us to verify certain cases of a conjecture of Sarnak and Xue.

LERF and the Lubotzky–Sarnak Conjecture

We prove that every closed hyperbolic 3‐manifold has a family of (possibly infinite sheeted) coverings with the property that the Cheeger constants in the family tend to zero. This is used to show that, if in addition the fundamental group of the manifold is LERF, then it satisfies the Lubotzky‐Sarnak conjecture. 57M50

A geometric characterization of arithmetic Fuchsian groups

The trace set of a Fuchsian group Γ encodes the set of lengths of closed geodesics in the surface Γ\H. Luo and Sarnak [3] showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering (BC) property. Sarnak [5] then conjectured that the BC property actually characterizes arithmetic Fuchsian groups. Schmutz [6] stated the even stronger conjecture that a cofinite Fuchsian group is arithmetic if its trace set has linear growth. He proposed a proof of this conjecture in the case when the group Γ contains at least one parabolic element, but unfortunately, this proof contains a gap. In this article, we point out this gap, and we prove Sarnak's conjecture under the assumption that the Fuchsian group Γ contains parabolic elements.

Compact unitary periods

Let $E$ be a $\textrm{CM}$ -field and $\pi$ a cuspidal representation of $\textrm{GL}_n({\mathbb{A}}_E)$ which admits a spherical vector (at all places) $\phi_0$ . We evaluate the period of $\phi_0$ with respect to any compact unitary group. The result is consistent with a conjecture of Sarnak.

Existence and Weyl’s law for spherical cusp forms

Let G be a split adjoint semisimple group over $${\user2{\mathbb{Q}}} $$ and $$ K _\infty \subset \mathbf{G} \user2{\mathbb{(R)}} $$ a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of $$ {\mathbf{G}}({\user2{\mathbb{R}}})/K_{\infty } $$ . This proves a conjecture of Sarnak for $$ \user2{\mathbb{Q}} $$ -split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula.

Weak Weyl's Law For Congruence Subgroups

A conjecture of Sarnak [Sa] states that the counting function fo the cuspidal spectrum satisfies Weyl's law. This conjecture has been established in some special cases. First of all, it was Selberg [Se] who proved it for congruence subgroups of $SL(2,Z)$ and $\sigma = 1$. Other cases for which the conjecture has been established are Hilbert modular groups [Ef], congruence subgroups of $SO(n,1)$ [Rez], $SL(3,Z)$ [Mil], and in particular, the conjecture was proved in [Mu2] for principal congruence subgroups of $SL(n,Z)$ and arbitrary $\sigma$ ... The purpose of this paper is to prove a weaker result which holds for every $G$. Recall that an upper bound, which holds for arbitrary $G$ and $\Gamma$, is already known thanks to Donnelly [Do] ... By working with a simple form of the trace formula, we shall get, for a general $G$, a lower bound that depends on the choice of a set $S$ of primes containing at least two finite primes. For every such set $S$ we shall define a certain constant $c_{S}(\Gamma}$ less than or equal to 1, which is non zero for $\Gamma$ deep enough.

On the rate of quantum ergodicity on hyperbolic surfaces and for billiards

The rate of quantum ergodicity is studied for three strongly chaotic (Anosov) systems. The quantal eigenfunctions on a compact Riemannian surface of genus g = 2 and of two triangular billiards on a surface of constant negative curvature are investigated. One of the triangular billiards belongs to the class of arithmetic systems. There are no peculiarities observed in the arithmetic systems concerning the rate of quantum ergodicity. This contrasts to the peculiar behaviour with respect to the statistical properties of the quantal levels. It is demonstrated that the rate of quantum ergodicity in the three considered systems fits well with the known upper and lower bounds. Furthermore, Sarnak's conjecture about quantum unique ergodicity for hyperbolic surfaces is confirmed numerically in these three systems.