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Substitution Delone sets. (English) Zbl 1037.52017
This fundamental article presents a theory about substitution Delone set families, i.e. families of Delone sets $$(X_1, \dots, X_n)$$ which satisfy an inflation functional equation. The authors study solutions to the inflation functional equation that are discrete and including solutions that do not correspond to tilings. They develop a structure theory, and these results are proved for general inflation functional equations, without primitivity restriction. They also study multiset solutions which correspond more closely to tilings.

##### MSC:
 52C23 Quasicrystals and aperiodic tilings in discrete geometry 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry)
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