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Undecidable translational tilings with only two tiles, or one nonabelian tile

Rachel Greenfeld, Terence Tao

Abstract

We construct an example of a group $G = \mathbb{Z}^2 \times G_0$ for a finite abelian group $G_0$, a subset $E$ of $G_0$, and two finite subsets $F_1,F_2$ of $G$, such that it is undecidable in ZFC whether $\mathbb{Z}^2\times E$ can be tiled by translations of $F_1,F_2$. In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of $E$ by the tiles $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $\mathbb{Z}^2$). A similar construction also applies for $G = \mathbb{Z}^d$ for sufficiently large $d$. If one allows the group $G_0$ to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile $F$. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.

Undecidable translational tilings with only two tiles, or one nonabelian tile

Abstract

We construct an example of a group for a finite abelian group , a subset of , and two finite subsets of , such that it is undecidable in ZFC whether can be tiled by translations of . In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of by the tiles , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in ). A similar construction also applies for for sufficiently large . If one allows the group to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile . The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.

Paper Structure

This paper contains 21 sections, 22 theorems, 228 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. A note on set-theoretic foundations
  3. Tilings by a single tile
  4. The periodic tiling conjecture
  5. Tilings by multiple tiles
  6. Main results
  7. Overview of proof
  8. Acknowledgments
  9. Notation
  10. Periodic tiling conjecture in one dimension
  11. Combining multiple tiling equations into a single equation
  12. From tiling to functions
  13. Reduction to the Hamming cube
  14. Reduction to systems of linear equations on boolean functions
  15. Reduction to a local Boolean constraint
  16. ...and 6 more sections

Key Result

Theorem 1.8

There exists an explicit finite abelian group $G_0$, a subset $E_0$ of $G_0$, and finite non-empty subsets $F_1,F_2$ of $\mathbb{Z}^2 \times G_0$ such that the tiling equation $\operatorname{Tile}(F_1,F_2; \mathbb{Z}^2 \times E_0)$ is undecidable (and hence aperiodic).

Figures (4)

  • Figure 1.1: The logical dependencies between the undecidability results in this paper (and in Ber). For each implication, there is listed either the section where the implication is proven, or the number of the key proposition or lemma that facilitates the implication. We also remark that Proposition \ref{["tile-function'"]} is proven using Lemma \ref{['rigid-tile']}, while Proposition \ref{['linear-encoding']} is proven using Corollary \ref{['tiling-system-2']}, which in turn follows from Lemma \ref{['tiling-system']}.
  • Figure 9.1: A tiling by the rigid tile $R$ constructed in Lemma \ref{['rigid-tile']}.
  • Figure 11.1: Maps between various subgroups (or subsets) of $S_{\mathbb{Z}_4^2}$ and $\mathbb{Z}_4^2$. Solid arrows denote group homomorphisms; hooked arrows denote injections; double-headed arrows denote surjections; and unlabeled hooked arrows denote inclusions.
  • Figure 11.2: Some of the sets and maps mentioned in Proposition \ref{['linear-encoding']}. (The notation is the same as in Figure \ref{['fig:group']}.)

Theorems & Definitions (58)

  • Remark 1.1
  • Definition 1.2: Undecidability and aperiodicity
  • Example 1.3
  • Remark 1.4
  • Conjecture 1.5: Periodic tiling conjecture
  • Definition 1.6: Undecidability and aperiodicity for multiple tiles
  • Remark 1.7
  • Theorem 1.8: Undecidable tiling equation with two tiles in $\mathbb{Z}^2\times G_0$
  • Theorem 1.9: Undecidable tiling equation with two tiles in $\mathbb{Z}^d$
  • Remark 1.10
  • ...and 48 more