Sarnak's Conjecture Research Articles

Minimal zero entropy subshifts can be unrestricted along any sparse set

Abstract We present a streamlined proof of a result essentially presented by the author in [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28(4) (2008), 1291–1322], namely that for every set $S = \{s_1, s_2, \ldots \} \subset \mathbb {N}$ of zero Banach density and finite set A, there exists a minimal zero-entropy subshift $(X, \sigma )$ so that for every sequence $u \in A^{\mathbb {Z}}$ , there is $x_u \in X$ with $x_u(s_n) = u(n)$ for all $n \in \mathbb {N}$ . Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the polynomial Sarnak conjecture reported by Eisner [A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris353(7) (2015), 569–572] which are significantly more general than some recently provided by Kanigowski, Lemańczyk and Radziwiłł [Prime number theorem for analytic skew products. Ann. of Math. (2)199 (2024), 591–705] and by Lian and Shi [A counter-example for polynomial version of Sarnak’s conjecture. Adv. Math.384 (2021), Paper no. 107765] and shows that no similar result can hold under only the assumptions of minimality and zero entropy.

Sarnak's conjecture in quantum computing, cyclotomic unitary group coranks, and Shimura curves

AbstractSarnak's conjecture in quantum computing concerns when the groups and over cyclotomic rings with , , are generated by the Clifford‐cyclotomic gate set. We previously settled this using Euler–Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group has only if is not generated by torsion elements. In this paper, we study the corank of these cyclotomic unitary groups in the families and , , by letting them act on Bruhat–Tits trees. The quotients by this action are finite graphs whose first Betti number is the corank of the group. Our main result is that for the families and , the corank grows doubly exponentially in as ; it is 0 precisely when , and indeed, the cyclotomic unitary groups are generated by torsion elements (in fact by Clifford‐cyclotomic gates) for these . We give explicit lower bounds for the corank in two different ways. The first is to bound the isotropy subgroups in the action on the tree by explicit cyclotomy. The second is to relate our graphs to Shimura curves over via interchanging local invariants and applying a result of Selberg and Zograf. We show that the cyclotomy arguments give the stronger bounds. In a final section, we execute a program of Sarnak to show that our results for the and families are sufficient to give a second proof of Sarnak's conjecture.

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.

On arithmetic functions orthogonal to deterministic sequences

We prove the equivalence of Sarnak's conjecture on Möbius orthogonality with a Kolmogorov type property conjectured by Veech for Furstenberg systems of the Möbius 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 characterization persisting in the logarithmic setup. As a corollary, we obtain that the logarithmic Sarnak's conjecture holds if and only if the logarithmic Möbius orthogonality is satisfied for all dynamical systems whose ergodic measures yield nilsystems.

Polynomial mean complexity and logarithmic Sarnak conjecture

AbstractIn 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.

Higher uniformity of bounded multiplicative functions in short intervals on average

Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$ \int^{2X}_X \sup_{\begin{smallmatrix} P(Y)\in\mathbb{R}[Y] \\ \mathrm{deg}{P} \leq k\end{smallmatrix}} \left| \sum_{x\leq n \leq x+H} \lambda(n) e(-P(n))\right| \ dx=o (XH) $$ for all fixed $k$ and $X^{\theta} \leq H \leq X$ with $0 \lt \theta \lt 1$ fixed but arbitrarily small. Previously this was only established for $k \leq 1$. We obtain this result as a special case of the corresponding statement for (non-pretentious) $1$-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases $e(-P(n))$ by degree $k$ nilsequences $\overline{F}(g(n) \Gamma )$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result $$ \int_{X}^{2X} \| \lambda \|_{U^{k+1}([x,x+H])}\ dx = o ( X ) $$ in the same range of $H$. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of $\lambda$ over short polynomial progressions $(n+P_1(m),\ldots , n+P_k(m))$, which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range $H\geq \mathrm{exp}((\mathrm{log} X)^{5/8+\varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.

Godement-Jacquet L-function, some conjectures and some consequences

In this paper, we investigate the mean square estimate for the logarithmic derivative of the Godement--Jacquet $L$-function $L_f(s)$ assuming the Riemann hypothesis for $L_f(s)$ and Rudnick--Sarnak conjecture.

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.

Sequence complexity, rigidity and logarithmic Sarnak conjecture

In this paper, we introduce sequence complexity and sequence mean complexity. It turns out that a topological dynamical system is uniformly rigid if and only if it has bounded topological sequence complexity; A measure preserving system is rigid if and only if it has bounded measure theoretic sequence complexity if and only if it has bounded measure theoretic sequence mean complexity. We obtain a formula of measure-theoretic entropy of an ergodic measure preserving system. We also prove that a topological dynamical systems $ (X, T) $ meets the Logarithmic Sarnak conjecture if each invariant measure of $ (X, T) $ has sub-linear measure theoretic sequence mean complexity.

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}.

One-Level Density and Non-Vanishing for Cubic L-Functions Over the Eisenstein Field

Abstract We study the one-level density for families of $L$-functions associated with cubic Dirichlet characters defined over the Eisenstein field. We show that the family of $L$-functions associated with the cubic residue symbols $\chi _n$ with $n$ square-free and congruent to 1 modulo 9 satisfies the Katz–Sarnak conjecture for all test functions whose Fourier transforms are supported in $(-13/11, 13/11)$, under the Generalized Riemann Hypothesis. This is the first result extending the support outside the trivial range$(-1, 1)$ for a family of cubic $L$-functions. This implies that a positive density of the $L$-functions associated with these characters do not vanish at the central point $s=1/2$. A key ingredient in our proof is a bound on an average of generalized cubic Gauss sums at prime arguments, whose proof is based on the work of Heath-Brown and Patterson [22, 23].

A counter-example for polynomial version of Sarnak's conjecture

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.

Sarnak’s conjecture for sequences of almost quadratic word growth

AbstractWe 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 words 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 ( $\kappa -1$ )-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\varepsilon })$ many words of length n where $t = \kappa (\kappa +1)/2$ . We prove a variant of the $1$ -Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension less than $1$ .

Möbius orthogonality for q-semimultiplicative sequences

We show that all q-semimultiplicative sequences are asymptotically orthogonal to the Mobius function, thus proving the Sarnak conjecture for this class of sequences. This generalises analogous results for the sum-of-digits function and other digital sequences which follow from previous work of Mauduit and Rivat.

Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows

We study time changes of bounded type Heisenberg nilflows (ϕ t ) acting on the Heisenberg nilmanifold M. We show that for every positive τ∈W s (M), s>7/2, every non-trivial time change (ϕ t τ ) enjoys the Ratner property. As a consequence, every mixing time change is mixing of all orders. Moreover, we show that for every τ∈W s (M), s>9/2 and every p,q∈ℕ, p≠q, (ϕ pt τ ) and (ϕ qt τ ) are disjoint. As a consequence, Sarnak conjecture on Möbius disjointness holds for all such time changes.

Orders of oscillation motivated by Sarnak’s conjecture

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 topological entropy and for all affine distal flows on the $d$-torus for all $d\geq 2$.

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 interval exchange transformations on three intervals

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.

A remark on Sarnak’s conjecture

We investigate Sarnak's conjecture on the Mobius function in the special case when the test function is the indicator of the set of integers for which a real additive function assumes a given value.

Prime points in orbits: Some instances of the Bourgain–Gamburd–Sarnak conjecture

We use Vaughan's variation on Vinogradov's three-primes theorem to prove Zariski-density of prime points in several infinite families of hypersurfaces, including level sets of some quadratic forms, the Permanent polynomial, and the defining polynomials of some pre-homogeneous vector spaces. Three of these families are instances of a conjecture by Bourgain, Gamburd and Sarnak regarding prime points in orbits of simple algebraic groups. Our approach is based on the formulation of a general condition on the defining polynomial of a hypersurface, which suffices to guarantee that Zariski-density of prime points is equivalent to the existence of an odd point.