Computing modular coincidences for substitution tilings and point sets

Frettlöh, Dirk and Sing, Bernd (2007). Computing modular coincidences for substitution tilings and point sets. Discrete and Computational Geometry, 37(3) pp. 381–401.

DOI: https://doi.org/10.1007/s00454-006-1280-9

URL: http://www.springerlink.com/content/1432-0444/

Abstract

Computing modular coincidences can show whether a given substitution system, which is supported on a point lattice in $R\!\!\!\! R^d$, consists of model sets or not. We prove the computatibility of this problem and determine an upper bound for the number of iterations needed. The main tool is a simple algorithm for computing modular coincidences, which is essentially a generalization of the Dekking coincidence to more than one dimension, and the proof of equivalence of this generalized Dekking coincidence and modular coincidence. As a consequence, we also obtain some conditions for the existence of modular coincidences. In a separate section, and throughout the article, a number of examples are given.

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