An application of topological  multiple recurrence to tiling

Rafael De La Llave; A. Windsor

doi:10.3934/dcdss.2009.2.315

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2009, Volume 2, Issue 2: 315-324. Doi: 10.3934/dcdss.2009.2.315

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An application of topological multiple recurrence to tiling

Rafael De La Llave^1, and
A. Windsor^2,

1.
Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712
2.
Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240

Received: March 31, 2008

Revised: July 31, 2008

Published: May 2009

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Abstract

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.

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Mathematics Subject Classification: 37B20, 37B50.

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