An Application of Topological Multiple Recurrence to Tiling

TL;DR

This paper applies topological multiple recurrence to tiling theory, demonstrating that large patches in any Euclidean tiling can approximate scaled and translated patterns, with a simple proof based on Furstenberg's theorem.

Contribution

It introduces a novel application of topological recurrence to tilings, providing a new method to find pattern approximations within large patches.

Findings

Large patches can approximate scaled and translated patterns in any Euclidean tiling.

The proof leverages Furstenberg's topological multiple recurrence theorem.

The approach simplifies understanding pattern distribution in tilings.

Abstract

We show that given any tiling of Euclidean space, any geometric patterns 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.