Coincidences and colorings of lattices and Z-modules [Elektronische Ressource] / Manuel Joseph C. Loquias. Fakultät für Mathematik. Forschungsschwerpunkt Mathematisierung (FSPM). Sonderforschungsbereich: Spektrale Strukturen und Topologische Methoden in der Mathematik (DFG SFB 701)

ZCoincidences and Colorings ofLattices and -modulesManuel Joseph C. LoquiasZCoincidences and Colorings ofLattices and -modulesDissertation zur Erlangung des Doktorgradesan derFakult¨at fu¨r MathematikUniversit¨at Bielefeldvorgelegt vonManuel Joseph C. LoquiasFakult¨at fu¨r MathematikUniversit¨at BielefeldNovember 2010ii1. Berichterstatter: Prof. Dr. Michael Baake und Dr. Peter Zeiner2. Berichterstatter: Prof. Dr. Uwe GrimmDatum der mu¨ndlichen Pru¨fung: 25. Januar 2011Pru¨fungsausschuss: Prof. Dr. M. Baake, Prof. Dr. W.-J. Beyn, Prof. Dr. H. Krause, Prof. Dr. M. Kaßmann,Dr. P. ZeinerGedruckt auf alterungsbest¨andigem Papier ISO 9706ZZZContentsAcknowledgements vIntroduction 1Background 1Outline of the thesis 2Chapter 1. Preliminaries 51.1. Coincidences of lattices and -modules 51.2. Solution of the coincidence problem for certain lattices and -modules 81.3. Lattice colorings 14Chapter 2. Coincidence indices of sublattices and colorings of lattices 152.1. Coincidence index with respect to a sublattice 152.2. Color coincidence 192.3. Further examples 232.4. Application to quasicrystals 28Chapter 3. Coincidences of shifted lattices 313.1. Affine coincidences 313.2. The coincidence problem for a shifted lattice 333.3. The setsH, AC(Γ), and OC(x+Γ) as groupoids 363.4. Coincidences of a shifted square lattice 363.5. Corresponding results for -modules 49Chapter 4. Coincidences of multilattices 534.1.