June  2009, 2(2): 315-324. doi: 10.3934/dcdss.2009.2.315

An application of topological multiple recurrence to tiling

1. 

Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, United States

2. 

Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240, United States

Received  April 2008 Revised  August 2008 Published  April 2009

We show that given any tiling of Euclidean space, any geometric pattern of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated version of the pattern. The rather simple proof uses Furstenberg's topological multiple recurrence theorem.
Citation: Rafael De La Llave, A. Windsor. An application of topological multiple recurrence to tiling. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 315-324. doi: 10.3934/dcdss.2009.2.315
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