Nick S
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Registered User
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May 7 |
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Mathematics of quasicrystals (ii) $\Lambda$ is relatively dense, uniformly discrete and $\Lambda-\Lambda \subset \Lambda+F$ for some finite set $F$..... It is somehow surprising, but one (hence both) of these conditions imply that $\supp(\mu_d)$ is relatively dense. In general Meyer sets have also non-trivial $\mu_c$. |
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May 7 |
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Mathematics of quasicrystals 2) While characterizing all point sets with these properties is hard, "geometric" conditions which imply each of them are known. Regular model sets are obtained from higher dimensional lattices and they always have $\mu_c=0$... Their subsets, Meyer sets can be characterized by one of the following two equivalent definitions (there are actually more, but I left the rest out): (i) $\Lambda$ is relatively dense and $\Lambda-\Lambda:=\{ x-y| x,y \in \Lambda\}$ is uniformly discrete. |
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May 7 |
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Mathematics of quasicrystals Just few comments: 1) $\mu_d \neq 0$ is not a good condition and I don't think Hof used it. The problem with it is that $\mu_d(\{ 0 \}) = (\dens (N) )^2$ so, unless N is trivial, $\mu_d \neq 0$. The usual two conditions we use are (i) $\mu_c =0$ or the weaker (ii) $\supp(\mu_d)$ is relatively dense. Also, in that paper Hof showed that thermal motion always induces some nontrivial $\mu_c$, thus $\mu_c=0$ is not really the right definition unless you work at 0K. |
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Apr 12 |
awarded | ● Nice Answer |
