Periodicity and decidability of translational tilings by rational polygonal sets

TL;DR

This paper proves the periodic tiling conjecture for rational polygonal sets in the plane, showing such tilings are always weakly periodic and the problem of determining tilings is decidable.

Contribution

It establishes the periodic tiling conjecture for rational polygons in 2D and proves the decidability of tilings by these sets, advancing understanding of tiling structures.

Findings

Periodic tiling conjecture holds for rational polygons in D.

Any tiling by a rational polygonal tile is weakly periodic.

Decidability of tilings by rational polygonal sets is proven.

Abstract

The periodic tiling conjecture asserts that if a region $\Sigma\subset \mathbb R^d$ tiles $\mathbb R^d$ by translations then it admits at least one fully periodic tiling. This conjecture is known to hold in $\mathbb R$, and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic tiling conjecture for polygonal sets: bounded open sets in $\mathbb R^2$ whose boundary is a finite union of line segments. We prove the periodic tiling conjecture for any polygonal tile whose vertices are rational. As a corollary of our argument, we also obtain the decidability of tilings by rational polygonal sets. Moreover, we prove that any translational tiling by a rational polygonal tile is weakly-periodic, i.e., can be partitioned into finitely many singly-periodic pieces.