An application of topological
multiple recurrence to tiling
Rafael De La Llave - Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, 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.
Received: April 2008; Revised: August 2008; Published: April 2009.