Undecidability of Translational Tiling with Three Tiles

TL;DR

This paper proves that determining translational tiling with three connected tiles in four-dimensional space is undecidable, supporting the conjecture that such tiling problems are undecidable in fixed dimensions.

Contribution

It provides a new undecidability result for translational tiling with three tiles in four dimensions, advancing understanding of tiling problem complexities.

Findings

Undecidability of 4D tiling with three connected tiles.

Supports conjecture of fixed-dimension tiling undecidability.

Builds on recent results disproving periodic tiling conjecture.

Abstract

Is there a fixed dimension $n$ such that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by showing that an aperiodic monotile exists in sufficiently high dimension $n$ [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension $n$ is part of the input, then the translational tiling for subsets of $\mathbb{Z}^n$ with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable, for some fixed $n$. This paper gives another supportive result for this conjecture by showing that translational tiling of the $4$-dimensional space with a set of three connected tiles is undecidable.