Doubly sparse measures on second countable locally compact Abelian groups are shown to be supported on finitely many translates of a lattice with trigonometric polynomial amplitudes.
Equicontinuous delone dynamical systems
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abstract
We characterize equicontinuous Delone dynamical systems as those coming from Delone sets with strongly almost periodic Dirac combs. Within the class of systems with nite local complexity the only equicontinuous systems are then shown to be the crystalline ones. On the other hand, within the class without nite local complexity, we exhibit examples of equicontinuous minimal Delone dynamical systems which are not crystalline. Our results solve the problem posed by Lagarias whether a Delone set whose Dirac comb is strongly almost periodic must be crystalline.
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math.MG 1years
2019 1verdicts
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Pure point measures with sparse support and sparse Fourier--Bohr support
Doubly sparse measures on second countable locally compact Abelian groups are shown to be supported on finitely many translates of a lattice with trigonometric polynomial amplitudes.