Lattices and multi-dimensional words

Authors:
Valérie Berthé;Robert Tijdeman
Affiliations:
LIRMM, 161 rue Ada, Montpellier F-34392, Cedex 05, France;Mathematical Institute, Leiden University, Postbus 9512, 2300 RA Leiden, The Netherlands
Venue:
Theoretical Computer Science - Combinatorics of the discrete plane and tilings
Year:
2004

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Abstract

In the present paper we develop a formalism to generate multi-dimensional words using lattices which generalizes the construction of real numbers (one-dimensional words) from a sequence of partial quotients using continued fractions. The construction was introduced in a special case by Simpson and Tijdeman in order to derive a multi-dimensional generalization of the theorem of Fine and Wilf. We show that the produced multi-dimensional words are intrinsically connected with k-dimensional Sturmian words.