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Perturbations of discrete lattices and almost periodic sets. (English) Zbl 1222.42011

Summary: A discrete set in the \(p\)-dimensional Euclidean space is almost periodic if the measure with the unit masses at points of the set is almost periodic in the weak sense. We propose to construct positive almost periodic discrete set as an almost periodic perturbation of a full rank discrete lattice. Also, we prove that each almost periodic discrete set on the real axes is an almost periodic perturbation of some arithmetic progression. Next, we consider signed almost periodic discrete sets, i.e., when the signed measure with masses \(+1\) or \(-1\) at points of a discrete set is almost periodic. We construct a signed discrete set that is not almost periodic, while the corresponding signed measure is almost periodic in the sense of distributions. Also, we construct a signed almost periodic discrete set such that the measure with masses \(+1\) at all points of the set is not almost periodic.

MSC:

42A75 Classical almost periodic functions, mean periodic functions
11K70 Harmonic analysis and almost periodicity in probabilistic number theory
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
52C23 Quasicrystals and aperiodic tilings in discrete geometry
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