Spectrum of a Rudin–Shapiro-like sequence

Chan, Lax; Grimm, Uwe

doi:10.1016/j.aam.2016.12.003

Cited by 15 publications

(9 citation statements)

References 13 publications

(24 reference statements)

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“…Another example that satisfies the √ L-criterion but evades AC spectral measures was analysed in [18]. Based on Bartlett's approach, it was shown to have singular continuous spectrum.…”

Section: Abelian Bijective Substitutionsmentioning

confidence: 99%

“…Our particular interest is the derivation of sufficient criteria for the absence of absolutely continuous diffraction and spectral measures, as well as necessary criteria for their presence. This is motivated by the rare occurrence of such components, which to date is essentially limited to Rudin-Shapiro-type sequences and their generalisations; compare [27,5,18,19] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

Baake

Gähler

Mañibo

2019

Commun. Math. Phys.

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The pair correlations of primitive inflation rules are analysed via their exact renormalisation relations. We introduce the inflation displacement algebra that is generated by the Fourier matrix of the inflation and deduce various consequences of its structure. Moreover, we derive a sufficient criterion for the absence of absolutely continuous diffraction components, as well as a necessary criterion for its presence. This is achieved via estimates for the Lyapunov exponents of the Fourier matrix cocycle of the inflation rule. While we develop the theory first for the classic setting in one dimension, we also present its extension to primitive inflation rules in higher dimensions with finitely many prototiles up to translations.

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“…Another example that satisfies the √ L-criterion but evades AC spectral measures was analysed in [18]. Based on Bartlett's approach, it was shown to have singular continuous spectrum.…”

Section: Abelian Bijective Substitutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

Baake

Gähler

Mañibo

2019

Commun. Math. Phys.

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“…an be of much larger order of magnitude than ? N for some z with |z| " 1, see [3, Theorem 2] as well as [16].…”

Section: Mauduit and Sárközymentioning

confidence: 99%

On the pseudorandomness of automatic sequences

Mérai

Winterhof

2017

Cryptogr. Commun.

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“…Diffraction image of the DPV (27). & Grimm (2017), Chan et al (2018) for examples of deterministic structures that show absolutely continuous diffraction]. Here, we briefly discuss a related concept that has recently gained popularity.…”

Section: Figure 11mentioning

confidence: 99%

Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction

Baake¹,

Grimm²

2020

Acta Cryst Sect A

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Tilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut-and-project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.

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Spectrum of a Rudin–Shapiro-like sequence

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Cited by 15 publications

References 13 publications

Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

On the pseudorandomness of automatic sequences

Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction

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