Pure point diffraction and cut and project schemes for measures: the smooth case

Lenz, Daniel; Richard, Christoph

doi:10.1007/s00209-006-0077-0

Cited by 54 publications

(128 citation statements)

References 39 publications

Supporting

Mentioning

124

Contrasting

Order By: Relevance

“…Dynamical systems techniques have turned out to be a powerful tool to analyse model sets [13,34,32,8,24]. Here one considers the hull of a model set, i.e., its translation orbit closure with respect to a Hausdorff-type metric, and one seeks to infer properties of the model set from its hull.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Periods and factors of weak model sets

Keller

Richard

2018

Isr. J. Math.

Self Cite

View full text Add to dashboard Cite

There is a renewed interest in weak model sets due to their connection to B-free systems [10], which emerged from Sarnak's program on the Möbius disjointness conjecture. Here we continue our recent investigation [22] of the extended hull M G W , a dynamical system naturally associated to a weak model set in an abelian group G with relatively compact window W. For windows having a nowhere dense boundary (this includes compact windows), we identify the maximal equicontinuous factor of M G W and give a sufficient condition when M G W is an almost 1-1 extension of its maximal equicontinuous factor. If the window is measurable with positive Haar measure and is almost compact, then the system M G W equipped with its Mirsky measure is isomorphic to its Kronecker factor. For general nontrivial ergodic probability measures on M G W , we provide a kind of lower bound for the Kronecker factor. All relevant factor systems are natural G-actions on quotient subgroups of the torus underlying the weak model set. These are obtained by factoring out suitable window periods. Our results are specialised to the usual hull of the weak model set, and they are also interpreted for B-free systems. * These notes profited enormously from the stimulating research atmosphere at the workshop "Combining Aperiodic Order with Structural Disorder" at the Lorentz Center, Leiden, 2016.

show abstract

Section: Introductionmentioning

confidence: 99%

“…There we also discuss applications of the results to B-free systems. A more detailed discussion about non-trivial topological behaviour in B- 2 The corresponding factor map ν G W was already systematically used for weighted model sets in [24], where it is continuous. free systems appears in [19].…”

Section: Introductionmentioning

confidence: 99%

Periods and factors of weak model sets

Keller

Richard

2018

Isr. J. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…As is well-known, see e.g. [33] for a recent discussion, pXpµq, Gq admits a group structure such that t Þ Ñ T t µ is a continuous group homomorphism if and only if µ is strongly almost periodic. Now, the statement follows from the equivalence of (i) and (ii) in (b) of Proposition 3.7.…”

Section: The Hull Of a Weakly Almost Periodic Measurementioning

confidence: 99%

On weakly almost periodic measures

Lenz

Strungaru

2019

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

We study the diffraction and dynamical properties of translation bounded weakly almost periodic measures. We prove that the dynamical hull of a weakly almost periodic measure is a weakly almost periodic dynamical system with unique minimal component given by the hull of the strongly almost periodic component of the measure. In particular the hull is minimal if and only if the measure is strongly almost periodic and the hull is always measurably conjugate to a torus and has pure point spectrum with continuous eigenfunctions. As an application we show the stability of the class of weighted Dirac combs with Meyer set or FLC support and deduce that such measures have either trivial or large pure point respectively continuous spectrum. We complement these results by investigating the Eberlein convolution of two weakly almost periodic measures. Here, we show that it is unique and a strongly almost periodic measure. We conclude by studying the Fourier-Bohr coefficients of weakly almost periodic measures.

show abstract

“…We will refer to these three components as the pure point, absolutely continuous and singular continuous diffraction spectra, respectively, of our structure. Structures with pure point diffraction, that is structures for which the absolutely and singular continuous diffraction spectra are absent, are now relatively well understood (see for example [6,9,10,11,12,23,25,24,27,28,30,31,44]). Of particular interest among pure point diffractive point sets are model sets, which are constructed by projecting points in a strip from a higher dimensional lattice (see Definition 2.8 below for the exact definition).…”

Section: Introductionmentioning

confidence: 99%

On the Fourier analysis of measures with Meyer set support

Strungaru

2020

Journal of Functional Analysis

View full text Add to dashboard Cite

In this paper we show the existence of the generalized Eberlein decomposition for Fourier transformable measures with Meyer set support. We prove that each of the three components is also Fourier transformable and has Meyer set support. We obtain that each of the pure point, absolutely continuous and singular continuous components of the Fourier transform is a strong almost periodic measure, and hence is either trivial or has relatively dense support. We next prove that the Fourier transform of a measure with Meyer set support is norm almost periodic, and hence so is each of the pure point, absolutely continuous and singular continuous components. We show that a measure with Meyer set support is Fourier transformable if and only if it is a linear combination of positive definite measures, which can be chosen with Meyer set support, solving a particular case of an open problem. We complete the paper by discussing some applications to the diffraction of weighted Dirac combs with Meyer set support.

show abstract

Pure point diffraction and cut and project schemes for measures: the smooth case

Cited by 54 publications

References 39 publications

Periods and factors of weak model sets

Periods and factors of weak model sets

On weakly almost periodic measures

On the Fourier analysis of measures with Meyer set support

Contact Info

Product

Resources

About