Ergodic Properties of Visible Lattice Points

Baake, Michael; Huck, Christian

doi:10.1134/s0371968515010136

Cited by 22 publications

(49 citation statements)

References 51 publications

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“…As we discuss in Subsection 2.2 below, the definition of a regular model set involves a number of technical conditions on the window W 0 . These condition can be weakened, leading to the notion of a weak model set [46,51,52], which has recently received a lot of attention in the abelian setting [3,4,[27][28][29][30][31]48]. It turns out that the algebraic structure of weak model sets is quite different from that of regular model sets.…”

Section: Model Setsmentioning

confidence: 99%

“…(1) to construct plenty of examples of mathematical quasi-crystals in non-abelian lcsc groups, and to point out some of the new phenomena which appear in this context; (2) to develop a theory of diffraction, which works in our general context, and specializes to the classical theory in the abelian case; (3) to compute in a rather explicit way the (spherical) diffraction of our examples.…”

Section: Introductionmentioning

confidence: 99%

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Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets

Björklund

Hartnick

Pogorzelski

2017

Proc. London Math. Soc.

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ABSTRACT. We introduce and study model sets in commutative spaces, i.e. homogeneous spaces of the form G/K where G is a (typically non-abelian) locally compact group and K is a compact subgroup such that (G, K) is a Gelfand pair. Examples include model sets in hyperbolic spaces, Riemannian symmetric spaces, regular trees and generalized Heisenberg groups. Continuing our work from [6] we associate with every regular model set in G/K a Radon measure on K\G/K called its spherical auto-correlation. We then define the spherical diffraction of the regular model set as the spherical Fourier transform of its spherical auto-correlation in the sense of Gelfand pairs. The main result of this article ensures that the spherical diffraction of a uniform regular model set in a commutative space is pure point. In fact, we provide an explicit formula for the spherical diffraction of such a model set in terms of the automorphic spectrum of the underlying lattice and the underlying window. To describe the coefficients appearing in this formula, we introduce a new type of integral transform for functions on the internal space of the model set. This integral transform can be seen as a shadow of the spherical Fourier transform of physical space in internal space and is hence referred to as the shadow transform of the model set. To illustrate our results we work out explicitly several examples, including the case of model sets in the Heisenberg group.

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Section: Model Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets

Björklund

Hartnick

Pogorzelski

2017

Proc. London Math. Soc.

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“…If by A we denote the set of arithmetic functions then pA,`,˚q is a ring which is an integral domain and the unit e P A is given by ½ t1u . 13 There is a natural ring isomorphism between A and the ring D of (formal) 14 Dirichlet series A Q u Þ Ñ U psq :" It is classical that if u and v are multiplicative then so is their Dirichlet convolution. The importance of multiplicativity can be seen in the representation of the Dirichlet series of a multiplicative function u as an Euler's product.…”

Section: Dirichlet Convolution Euler's Productmentioning

confidence: 99%

“…Here Λ is a lattice in R d and B Ď Nzt1u is an infinite pairwise coprime set with ř bPB 1{b d ă 8. • Finally, one can consider B-free integers F B in number fields as suggested in [13]. Here K is a finite extension of Q, O K Ă K is the ring of integers and B is a family of pairwise coprime ideals in O K such that the sum of reciprocals of their norms converges.…”

Section: Introductionmentioning

confidence: 99%

Sarnak’s Conjecture: What’s New

Ferenczi

Kułaga-Przymus

Lemańczyk

2018

Lecture Notes in Mathematics

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An overview of last seven years results concerning Sarnak's conjecture on Möbius disjointness is presented, focusing on ergodic theory aspects of the conjecture.1 Most often, however not always, T will be a homeomorphism. 2 µ stands for the arithmetic Möbius function, see next sections for explanations of notions that appear in Introduction.3 To be compared with Möbius Randomness Law by Iwaniec and Kowalski [95], page 338, that any "reasonable" sequence of complex numbers is orthogonal to µ.

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“…The first one is proved in Proposition 3.3. 2 A sequence of abelian groups and homomorphisms ...−→M k−1…”

Section: The Maximal Equicontinuous Factormentioning

confidence: 99%

Dynamics of $\boldsymbol{{\mathcal B}}$-Free Sets: A View Through the Window

Kasjan

Keller

Lemańczyk

2017

International Mathematics Research Notices

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Let B be an infinite subset of {1, 2, . . . }. We characterize arithmetic and dynamical properties of the B-free set F B through group theoretical, topological and measure theoretic properties of a set W (called the window) associated with B. This point of view stems from the interpretation of the set F B as a weak model set. Our main results are: B is taut if and only if the window is Haar regular; the dynamical system associated to F B is a Toeplitz system if and only if the window is topologically regular; the dynamical system associated to F B is proximal if and only if the window has empty interior; and the dynamical system associated to F B has the "naïvely expected" maximal equicontinuous factor if and only if the interior of the window is aperiodic.Therefore the implication (i) ⇒ (iii) of Theorem A is an immediate consequence of the following proposition. Proposition 1.1. Assume that B is taut. Let h ∈ W and S ⊂ B finite. Then the set of B-free integers n that solve n = h b mod b for b ∈ S has asymptotic density m H (U S (h) ∩ W) > 0.In Subsection 2.4 we provide a sequence B, which is not taut, but for which ∆( ) ∩ W = W (Example 2.2). Hence (iii) of Theorem A is not equivalent to (i) and (ii). Here we provide two simpler examples which throw some light on property (iii). Denote by P ⊆ AE the set of all prime numbers. Example 1.1. If B = P then H = p∈P /p , W is uncountable (although of Haar measure zero) and ∆( ) ∩ W W, since for each n we find p ∈ P such that p | n, so n = 0 mod p. Example 1.2. If B ⊂ P is thin, i.e. if p∈B 1/p < +∞, then ∆( ) ∩ W = W in view of (4), because each h ∈ H satisfies the B-free CRT. Indeed, if S ⊂ B is finite and n = h b mod b for b ∈ S , then n + lcm(S ) is the set of all solutions to this system of congruences. Moreover, if h ∈ W, then gcd(n, b) = 1 for all b ∈ S . We only need to find r ∈ so that n + r lcm(S ) is a prime number which is not in B. The latter follows from Dirichlet's theorem: The set of prime numbers contained in n + lcm(S ) is not thin. Of course this is a special case of Theorem A. Remark 1.2. Denote by ν η := m H • ϕ −1 the Mirsky measure on X η . There are two independent proofs of the fact that the two equivalent conditions from Theorem A imply that the measure preserving dynamical system (X η , σ, ν η ) is isomorphic to the group rotation (H, R ∆(1) , m H ): In [4, Theorem F] it is proved that this is implied by (i). That it is also a direct consequence of (ii) follows -in the more

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Ergodic Properties of Visible Lattice Points

Cited by 22 publications

References 51 publications

Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets

Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets

Sarnak’s Conjecture: What’s New

Dynamics of $\boldsymbol{{\mathcal B}}$-Free Sets: A View Through the Window

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