An application of topological multiple recurrence to tiling

Pages: 315 - 324, Volume 2, Issue 2, June 2009

doi:10.3934/dcdss.2009.2.315       Abstract        Full Text (225.3K)       Related Articles

Rafael De La Llave - Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, United States (email)
A. Windsor - Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240, United States (email)

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.

Keywords:  Tiling, Pattern, recurrence, multiple topological recurrence.
Mathematics Subject Classification:  37B20, 37B50.

Received: April 2008;      Revised: August 2008;      Published: April 2009.