Modulated crystals and almost periodic measures

Abstract

Modulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyze these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similar to de Bruijn’s analysis, we find stability with respect to almost periodic modulations.

Abstract almost periodicity for group actions on uniform topological spaces

“…Indeed, if μ ∈ M ∞ (G), then all measures in H Up μ are equi-translation bounded, and hence this orbit is complete. Therefore, we get the standard characterization of SAP(G), the first three equivalences appearing in [1,8,20], whereas the equivalence to condition (iv) appearing in [12,13,16]. Theorem 6.…”

“…Group valued almost periodic functions. In this section we review the concept of group valued almost periodic functions, as discussed recently in [12].…”

“…In this section, we review the concept of group-valued almost periodic functions, as discussed recently in [12]. Let G, H be two LCAG, with H complete.…”

Abstract almost periodicity for group actions on uniform topological spaces

“…Group valued almost periodic functions. In this section we review the concept of group valued almost periodic functions, as discussed recently in [12]. Let G, H be two LCAG, with H complete.…”

“…In many situations, the Bohr and Bochner definition of almost periodicity are equivalent (see for example [10,8,19,16] just to name a few). Moreover, the hull of an almost periodic function/measure is often a compact Abelian group [13,15,12,19]. Since the proofs in these related but different situations are similar, it is natural to ask if there may be any unified theory of almost periodicity which shows the equivalence between Bohr and Bochner definition in a very general situation.…”