Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

We construct an example of a group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = \mathbb {Z}^2 \times G_0$$\end{document}G=Z2×G0 for a finite abelian group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_0$$\end{document}G0, a subset E of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_0$$\end{document}G0, and two finite subsets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1,F_2$$\end{document}F1,F2 of G, such that it is undecidable in ZFC whether \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^2\times E$$\end{document}Z2×E can be tiled by translations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1,F_2$$\end{document}F1,F2. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1,F_2$$\end{document}F1,F2, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}^2$$\end{document}Z2). A similar construction also applies for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=\mathbb {Z}^d$$\end{document}G=Zd for sufficiently large d. If one allows the group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_0$$\end{document}G0 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.

1. Introduction 1.1.A note on set-theoretic foundations.In this paper we will be discussing questions of decidability in the Zermelo-Frankel-Choice (ZFC) axiom system of set theory.As such, we will sometimes have to make distinctions between the standard universe 1 U of ZFC, in which for instance the natural numbers N = N U are the standard natural numbers {0, 1, 2, . . .}, the integers Z = Z U are the standard integers {0, ±1, ±2, . . .}, and so forth, and also nonstandard universes U * of ZFC, in which the model N U * of the natural numbers may possibly admit some nonstandard elements not contained in the standard natural numbers N U , and similarly for the model Z U * of the integers in this universe.However, every standard natural number n = n U ∈ N will have a well-defined counterpart n U * ∈ N U * in such universes, which by abuse of notation we shall usually identify with n; similarly for standard integers.
If S is a first-order sentence in ZFC, we say that S is (logically) undecidable (or independent of ZFC ) if it cannot be proven within the axiom system of ZFC.By the Gödel completeness theorem, this is equivalent to S being true in some universes of ZFC while being false in others.For instance, if S is a undecidable sentence that involves the group Z d for some standard natural number d, it could be that S holds for the standard model Z d = Z d U of this group, but fails for some non-standard model Z d U * of the group.
Remark 1.1.In the literature the closely related concept of algorithmic undecidability from computability theory is often used.By a problem S(x), x ∈ X we mean a sentence S(x) involving a parameter x in some range X that can be encoded as a binary string.Such a problem is algorithmically undecidable if there is no Turing machine T which, when given x ∈ X (encoded as a binary string) as input, computes the truth value of S(x) (in the standard universe) in finite time.One relation between the two concepts is that if the problem S(x), x ∈ X is algorithmically undecidable then there must be at least one instance S(x 0 ) of this problem with x 0 ∈ X that is logically undecidable, since otherwise one could evaluate the truth value of a sentence S(x) for any x ∈ X by running an algorithm to search for proofs or disproofs of S(x).Our main results on logical undecidability can also be modified to give (slightly stronger) algorithmic undecidability results; see Remark 1.12 below.However, we have chosen to use the language of logical undecidability here rather than algorithmic undecidability, as the former concept can be meaningfully applied to individual tiling equations, rather than a tiling problem involving one or more parameters x.
In order to describe various mathematical assertions as first-order sentences in ZFC, it will be necessary to have the various parameters of these assertions presented in a suitably "explicit" or "definable" fashion.In this paper, this will be a particular issue with regards to finitely generated abelian groups G = (G, +).Define an explicit finitely generated abelian group to be a group of the form for some (standard) natural numbers d, m and (standard) positive integers N 1 , . . ., N m , where we use Z N := Z/N Z to denote the standard cyclic group of order N .For instance, Z2 × Z 20 21 is an explicit finitely generated abelian group.We define the notion of a explicit finite abelian group similarly by omitting the Z d factor.From the classification of finitely generated abelian groups, we know that (in the standard universe U of ZFC) every finitely generated abelian group is (abstractly) isomorphic to an explicit finitely generated abelian group, but the advantage of working with explicit finitely generated abelian groups is that such groups G are definable in ZFC, and in particular have counterparts G U * in all universes U * of ZFC, not just the standard universe U.
1.2.Tilings by a single tile.If G is an abelian group and A, F are subsets of G, we define the set A ⊕ F to be the set of all sums a + f with a ∈ A, f ∈ F if all these sums are distinct, and leave A ⊕ F undefined if the sums are not distinct.Note that from our conventions we have A ⊕ F = ∅ whenever one of A, F is empty.Given two sets F, E in G, we let Tile(F ; E) denote the tiling equation 2 where we view the tile F and the set E to be tiled as given data and the indeterminate variable X denotes an unknown subset of G.We will be interested in the question of whether this tiling equation Tile(F ; E) admits solutions X = A, and more generally what the space Tile(F ; E) U := {A ⊂ G : A ⊕ F = E} of solutions to Tile(F ; E) looks like.Later on we will generalize this situation by considering systems of tiling equations rather than just a single tiling equation, and also allow for multiple tiles F 1 , . . ., F J rather than a single tile F .
We will focus on tiling equations in which G is a finitely generated abelian group, F is a finite subset of G, and E is a subset of G which is periodic, by which we mean3 that E is a finite union of cosets of some finite index subgroup of G.In order to be able to talk about the decidability of such tiling problems we will need to restrict further by requiring that G is an explicit finitely generated abelian group in the sense (1.1) discussed previously.The finite set F can then be described explicitly in terms of a finite number of standard integers; for instance, if F is a finite subset of Z 2 × Z N , then one can write it as for some standard natural number k and some standard integers a 1 , . . ., a k , b 1 , . . ., b k , c 1 , . . ., c k .Thus F is now a definable set in ZFC and has counterparts F U * in every universe U * of ZFC.Similarly, a periodic subset E of an explicit finitely generated abelian group Z d × Z N 1 × • • • × Z Nm can be written as for some standard natural number r and some finite subset S of G; thus E is also definable and has counterparts E U * in every universe U * of ZFC.One can now consider the solution space Tile(F ; E) U * := {A ⊂ G U * : A ⊕ F U * = E U * } to Tile(F ; E) in any universe U * of ZFC.
We now consider the following two properties of the tiling equation Tile(F ; E).Definition 1.2 (Undecidability and aperiodicity).Let G be an (explicit) finitely generated abelian group, F a finite subset of G, and E a periodic subset of G.
(i) We say that the tiling equation Tile(F ; E) is undecidable if the assertion that there exists a solution A ⊂ G to Tile(F ; E), when phrased as a firstorder sentence in ZFC, is not provable within the axiom system of ZFC.By the Gödel completeness theorem, this is equivalent to the assertion that Tile(F ; E) U * is empty for some4 universes U * of ZFC, but non-empty for some other universes.We say that the tiling equation Tile(F ; E) is decidable if it is not undecidable.
(ii) We say that the tiling equation Tile(F ; E) is aperiodic if, when working within the standard universe U of ZFC, the equation Tile(F ; E) admits a solution A ⊂ G, but that none of these solutions are periodic.That is to say, Tile(F ; E) U is non-empty but contains no periodic sets.
Example 1.3.Let G be the explicit finitely generated abelian group G := Z 2 , let F := {0, 1} 2 , and let E := Z 2 .The tiling equation Tile(F ; E) has multiple solutions in the standard universe U of ZFC; for instance, given any (standard) function a : Z → {0, 1}, the set solves the tiling equation Tile(F ; E) and is thus an element of Tile(F ; E) U .
Most of these solutions will not be periodic, but for instance if one selects the function a ≡ 0 (so that A = (2Z) 2 ) then one obtains a periodic tiling.This latter tiling is definable and thus has a counterpart in every universe U * of ZFC, and we conclude that in this case the tiling equation Tile(F ; E) is decidable and not aperiodic.
Remark 1.4.The notion of aperiodicity of a tiling equation Tile(F ; E) is only interesting when E is itself periodic, since if A ⊕ F = E and A is periodic then E must necessarily be periodic also.
A well-known argument of Wang (see [B66, R71]) shows that if a tiling equation Tile(F ; E) is not aperiodic, then it is decidable; contrapositively, if a tiling equation is undecidable, then it must also be aperiodic.From this we see that any undecidable tiling equation must admit (necessarily non-periodic) solutions in the standard universe of ZFC (because the tiling equation is aperiodic), but (by the completeness theorem) will not admit solutions at all in some other (nonstandard) universes of ZFC.For the convenience of the reader we review the proof of this assertion (generalized to multiple tiles, and to arbitrary periodic subsets E of explicit finitely generated abelian groups G) in Appendix A.
1.3.The periodic tiling conjecture.The following conjecture was proposed in the case5 E = G = Z d by Lagarias and Wang [LW96] and also previously appears implicitly in [GS87,p. 23]: Conjecture 1.5 (Periodic tiling conjecture).Let G be an explicit finitely generated abelian group, let F be a finite non-empty subset of G, and let E be a periodic subset of G. Then Tile(F ; E) is not aperiodic.
By the previous discussion, Conjecture 1.5 implies that the tiling equation Tile(F ; E) is decidable for every F, E, G obeying the hypotheses of the conjecture.
The following progress is known towards the periodic tiling conjecture: • Conjecture 1.5 is trivial when G is a finite abelian group, since in this case all subsets of G are periodic.
• When E = G = Z, Conjecture 1.5 was established by Newman [N77] as a consequence of the pigeonhole principle.In fact, the argument shows that every set in Tile(F ; Z) U is periodic.As we shall review in Section 2 below, the argument also extends to the case G = Z × G 0 for an (explicit) finite abelian group G 0 , and to an arbitrary periodic subset E of G. See also the results in Section 10 for some additional properties of one-dimensional tilings.
• When E = G = Z 2 , Conjecture 1.5 was established by Bhattacharya [B20] using ergodic theory methods (viewing Tile(F ; Z 2 ) U as a dynamical system using the translation action of Z 2 ).In our previous paper [GT20] we gave an alternative proof of this result, and generalized it to the case where E is a periodic subset of G = Z 2 .In fact, we strengthen the previous result of Bhattacharya, by showing that every set in Tile(F, E) U is weakly periodic (a disjoint union of finitely many one-periodic sets).In the case of polyominoes (where F is viewed as a union of unit squares whose boundary is a simple closed curve), the conjecture was previously established in [BN91], [G-BN91]6 and decidability was established even earlier in [WvL84].
The conjecture remains open in other cases; for instance, the case E = G = Z 3 or the case E = G = Z 2 × Z N for an arbitrary natural number N , are currently unresolved, although we hope to report on some results in these cases in forthcoming work.In [S98] it was shown that Conjecture 1.5 for E = G = Z d was true whenever the cardinality |F | of F was prime, or less than or equal to four.
1.4.Tilings by multiple tiles.It is natural to ask if Conjecture 1.5 extends to tilings by multiple tiles.Given subsets F 1 , . . ., F J , E of a group G, we use Tile(F 1 , . . ., F J ; E) = Tile((F j ) J j=1 ; E) to denote the tiling equation7 where A ⊎ B denotes the disjoint union of A and B (equal to A ∪ B when A, B are disjoint, and undefined otherwise).As before we view F 1 , . . ., F J , E as given data for this equation, and X 1 , . . ., X J are indeterminate variables representing unknown tiling sets in G.If G is an explicit finitely generated group, F 1 , . . ., F J are finite subsets of G, and E is a periodic subset of G, we can define the solution set and more generally for any other universe U * of ZFC we have We extend Definition 1.2 to multiple tilings in the natural fashion: Definition 1.6 (Undecidability and aperiodicity for multiple tiles).Let G be an explicit finitely generated abelian group, F 1 , . . ., F J be finite subsets of G for some standard natural number J, and E a periodic subset of G.
(i) We say that the tiling equation Tile(F 1 , . . ., F J ; E) is undecidable if the assertion that there exist subsets A 1 , . . ., A J ⊂ G solving Tile(F 1 , . . ., F J ; E), when phrased as a first-order sentence in ZFC, is not provable within the axiom system of ZFC.By the Gödel completeness theorem, this is equivalent to the assertion that Tile(F 1 , . . ., F J ; E) U * is non-empty for some universes U * of ZFC, but empty for some other universes.We say that Tile(F 1 , . . ., F J ; E) is decidable if it is not undecidable.(ii) We say that the tiling equation Tile(F 1 , . . ., F J ; E) is aperiodic if, when working within the standard universe U of ZFC, the equation Tile(F 1 , . . ., F J ; E) admits a solution A 1 , . . ., A J ⊂ G, but there are no solutions for which all of the A 1 , . . ., A J are periodic.That is to say, Tile(F 1 , . . ., F J ; E) U is non-empty but contains no tuples of periodic sets.
As in the single tile case, undecidability implies aperiodicity; see Appendix A. The argument of Newman that resolves the one-dimensional case of Conjecture 1.5 also shows that for (explicit) one-dimensional groups G = Z × G 0 , every tiling equation Tile(F 1 , . . ., F J ; E) is not aperiodic (and thus also decidable); see Section 2.
However, in marked contrast to what Conjecture 1.5 predicts to be the case for single tiles, it is known that a tiling equation Tile(F 1 , . . ., F J ; E) can be aperiodic or even undecidable when J is large enough.In the model case E = G = Z 2 , an aperiodic tiling equation Tile(F 1 , . . ., F J ; Z 2 ) was famously constructed8 by Berger [B66] with J = 20426, and an undecidable tiling was also constructed by a modification of the method with an unspecified value of J.A simplified proof of this latter fact was given by Robinson [R71], who also constructed a collection of J = 36 tiles was constructed in which a related completion problem was shown to be undecidable.The value of J for either undecidable examples or aperiodic examples has been steadily lowered over time; see Table 1 for a partial list.We refer the reader to the recent survey [JV20] for more details of these results.To our knowledge, the smallest known value of J for an aperiodic tiling equation Tile(F 1 , . . ., F J ; Z 2 ) is J = 8, by Ammann, Grünbaum,and Shephard [AGS92].The smallest known value of J for a tiling equation Tile(F 1 , . . ., F J ; Z 2 ) that was explicitly constructed and shown to be undecidable is J = 11, due to Ollinger [O09].
Remark 1.7.As Table 1 demonstrates, many of these constructions were based on a variant of a tile set in Z 2 known as a set of Wang tiles, but in [JR21] it was shown that Wang tile constructions cannot create aperiodic (or undecidable) tile sets for any J < 11.
1.5.Main results.Our first main result is that one can in fact obtain undecidable (and hence aperiodic) tiling equations for J as small as 2, at the cost of enlarging E from Z 2 to Z 2 × E 0 for some subset E 0 of a (explicit) finite abelian group G .
Theorem 1.8 (Undecidable tiling equation with two tiles in Z 2 × G 0 ).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 Z 2 × G 0 such that the tiling equation Tile(F 1 , F 2 ; Z 2 × E 0 ) is undecidable (and hence aperiodic).
The proof of Theorem 1.8 goes on throughout Sections 3-8.In Section 9, by "pulling back" the proof of Theorem 1.8, we prove the following analogue in Z d .Theorem 1.9 (Undecidable tiling equation with two tiles in Z d ).There exists an explicit d > 1, a periodic subset E of Z d , and finite non-empty subsets F 1 , F 2 of Z d such that the tiling equation Tile(F 1 , F 2 ; E) is undecidable (and hence aperiodic).
Remark 1.10.One can further extend our construction in Theorem 1.9 to the Euclidean space R d , as follows.First, replace each tile F j ⊂ Z d , j = 1, 2, with a finite union Fj of unit cubes centered in F j , and similarly replace E ⊂ Z d with a periodic set Ẽ ⊂ R d .Next, in order to make the construction rigid in the Euclidean space, add "bumps" on the sides (as in the proof of Lemma 9.3).When one does so, the only tilings of Ẽ by the F1 , F2 arise from tilings of E by F 1 , F 2 , possibly after applying a translation, and hence the undecidability of the former tiling problem is equivalent to that of the latter.
Our construction can in principle give a completely explicit description of the sets G 0 , E 0 , F 1 , F 2 , but they are quite complicated (and the group G 0 is large), and we have not attempted to optimize the size and complexity of these sets in order to keep the argument as conceptual as possible.
Remark 1.11.Our argument establishes an encoding for any tiling problem Tile(F 1 , . . ., F J ; Z 2 ) with arbitrary number of tiles in Z 2 as a tiling problem with two tiles in Z 2 × G 0 .However, in order to prove Theorem 1.8 we only need to be able to encode Wang tilings.
Remark 1.12.A slight modification of the proof of Theorem 1.8 also establishes the slightly stronger claim that the decision problem of whether the tiling equation Tile(F 1 , F 2 ; Z 2 × E 0 ) is solvable for a given finite abelian group G 0 , given finite non-empty subsets F 1 , F 2 ⊂ Z 2 × G 0 and E 0 ⊂ G 0 , is algorithmically undecidable.Similarly for Theorems 1.9, 11.2 below.This is basically because the original undecidability result of Berger [B66] that we rely on is also phrased in the language of algorithmic undecidability; see Footnote 11 in Section 8. We leave the details of the appropriate modification of the arguments in the context of algorithmic decidability to the interested reader.
Theorem 1.8 supports the belief9 that the tiling problem is considerably less well behaved for J ⩾ 2 than it is for J = 1.As another instance of this belief, the J = 1 tilings enjoy a dilation symmetry (see [B20,Proposition 3.1], [GT20, Lemma 3.1], [T95]) that have no known analogue for J ⩾ 2. We present a further distinction between the J = 1 and J ⩾ 2 situations in Section 10 below, where we show that in one dimension the J = 1 tilings exhibit a certain Thm.1.17 Thm.9.5 Thm. 1.16 Thm.9.2 Thm. 1.8 Thm.1.9 §8 §7 §7 The logical dependencies between the undecidability results in this paper (and in [B66]).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 9.4 is proven using Lemma 9.3, while Proposition 11.6 is proven using Corollary 11.5, which in turn follows from Lemma 11.4.
partial rigidity property that is not present in the J ⩾ 2 setting, and makes any attempt to extend our methods of proof of Theorem 1.8 to the J = 1 case difficult.On the other hand, if one allows the group G 0 to be nonabelian, then we can reduce the two tiles in Theorem 1.8 to a single tile: see Section 11.
1.6.Overview of proof.We now discuss the proof of Theorem 1.8; the proofs of Theorems 1.9, 11.2 are proven by modifications of the method and are discussed in Sections 9, 11 respectively.
The arguments proceed by a series of reductions in which we successively replace the tiling equation (1.2) by a more tractable system of equations; see Figure 1.1.
We first extend Definition 1.6 to systems of tiling equations.
Definition 1.13 (Undecidability and aperiodicity for systems of tiling equations with multiple tiles).Let G be an explicit finitely generated abelian group, J, M ⩾ 1 be standard natural numbers, and for each m = 1, . . ., M , let be finite subsets of G, and let E (m) be a periodic subset of G.
(i) We say that the system Tile(F J ; E (m) ) for all m = 1, . . ., M , when phrased as a first-order sentence in ZFC, is not provable within the axiom system of ZFC.That is to say, the solution set is non-empty in some universes U * of ZFC, and empty in others.We say that the system is decidable if it is not undecidable.(ii) We say that the system Tile(F when working within the standard universe U of ZFC, this system admits a solution A 1 , . . ., A J ⊂ G, but there are no solutions for which all of the A 1 , . . ., A J are periodic.That is to say, the solution set is non-empty but contains no tuples of periodic sets. Example 1.14.Let G be an explicit finitely generated abelian group, and let G 0 be a explicit finite abelian group.The solutions A to the tiling equation Tile({0} × G 0 ; G × G 0 ) are precisely those sets which are graphs for an arbitrary function f : G → G 0 .It is possible to impose additional conditions on f by adding more tiling equations to this "base" tiling equation Tile({0} × G 0 ; G × G 0 ).For instance, if in addition H is a subgroup of G 0 and y + H is a coset of H in G 0 , solutions A to the system of tiling equations are precisely sets A of the form (1.4) where the function f obeys the additional10 constraint f (n) ∈ y + H for all n ∈ G.As a further example, if −y 0 , y 0 are distinct elements of G 0 , and h is a non-zero element of G, then solutions A to the system of tiling equations are precisely sets A of the form (1.4) where the function f takes values in {−y 0 , y 0 } and obeys the additional constraint In all three cases one can verify that the system of tiling equations is decidable and not aperiodic.

We then have
Theorem 1.15 (Combining multiple tiling equations into a single equation).
Let J, M ⩾ 1, let G = Z d × G 0 be an explicit finitely generated abelian group for some explicit finite abelian group G 0 .Let Z N be a cyclic group with N > M , and for each m = 1, . . ., M let and Theorem 1.15 can be established by easy elementary considerations; see Section 3. In view of this theorem, Theorem 1.8 now reduces to the following statement.
Theorem 1.16 (An undecidable system of tiling equations with two tiles in Z 2 × G 0 ).There exists an explicit finite abelian group G 0 , a standard natural number M , and for each m = 1, . . ., M there exist finite non-empty sets ⊂ G 0 such that the system of tiling equations Tile(F The ability to now impose an arbitrary number of tiling equations grants us a substantial amount of flexibility.In Section 4 we will take advantage of this flexibility to replace the system of tiling equations with a system of functional equations, basically by generalizing the constructions provided in Example 1.14.Specifically, we will reduce Theorem 1.16 to the following statement. Theorem 1.17 (An undecidable system of functional equations).There exists an explicit finite abelian group G 0 , a standard integer M ⩾ 1, and for each m = 1, . . ., M there exist (possibly empty) finite subsets for all n ∈ Z 2 × Z 2 and m = 1, . . ., M is undecidable (when expressed as a first-order sentence in ZFC).
In the above theorem, the functions f 1 , f 2 can range freely in the finite group G 0 .By taking advantage of the Z 2 factor in the domain, we can restrict f 1 , f 2 to range instead in a Hamming cube {−1, 1} D ⊂ Z D N , which will be more convenient for us to work with, at the cost of introducing an additional sign in the functional equation (1.7).More precisely, in Section 5 we reduce Theorem 1.17 to Theorem 1.18 (An undecidable system of functional equations in the Hamming cube).There exist standard integers N > 2 and D, M ⩾ 1, and for each m = 1, . . ., M there exist shifts h ∈ Z 2 and (possibly empty sets) N for m = 1, . . ., M such that the question of whether there exist functions f 1 , f 2 : Z 2 → {−1, 1} D that solve the system of functional equations for all n ∈ Z 2 , m = 1, . . ., M , and ϵ = ±1 is undecidable (when expressed as a first-order sentence in ZFC).
The next step is to replace the functional equations (1.8) with linear equations on Boolean functions f j,d : Z 2 → {−1, 1} (where we now view {−1, 1} as a subset of the integers).More precisely, in Section 6 we reduce Theorem 1.18 to Theorem 1.19 (An undecidable system of linear equations for Boolean functions).There exist standard integers for all n ∈ Z 2 , j = 1, 2, and m = 1, . . ., M j , as well as the system of linear functional equations for all n ∈ Z 2 and d = 1, . . ., D 0 , is undecidable (when expressed as a first-order sentence in ZFC).
Now that we are working with linear equations for Boolean functions, we can encode a powerful class of constraints, namely all local Boolean constraints.In Section 7 we will reduce Theorem 1.19 to Theorem 1.20 (An undecidable local Boolean constraint).There exist standard integers D, L ⩾ 1, shifts h 1 , . . ., h L ∈ Z 2 , and a set Ω ⊂ {−1, 1} DL such that the question of whether there exist functions for all n ∈ Z 2 is undecidable (when expressed as a first-order sentence in ZFC).
Finally, in Section 8 we use the previously established existence of undecidable translational tile sets to prove Theorem 1.20, and thus Theorem 1.8.1.7.Acknowledgments.RG was partially supported by the Eric and Wendy Schmidt Postdoctoral Award.TT was partially supported by NSF grant DMS-1764034 and by a Simons Investigator Award.We gratefully acknowledge the hospitality and support of the Hausdorff Institute for Mathematics where a significant portion of this research was conducted.
We thank David Roberts for drawing our attention to the reference [ST11], Hunter Spink for drawing our attention to the reference [GLT16], Jarkko Kari for drawing our attention to the references [K92, KP99, L09], and Zachary Hunter and Matthew Foreman for further corrections.We are also grateful to the anonymous referee for several suggestions that improved the exposition of this paper.
1.8.Notation.Given a subset A ⊂ G of an abelian group G and a shift h ∈ G, we define The unary operator − is understood to take precedence over the binary operator ×, which in turn takes precedence over the binary operator ⊕, which takes precedence over the binary operator ⊎.Thus for instance By slight abuse of notation, any set of integers will be identified with the corresponding set of residue classes in a cyclic group Z N , if these classes are distinct.For instance, if M ⩽ N , we identify {1, . . ., M } with the residue classes {1 mod N, . . ., M mod N } ⊂ Z N , and if N > 2, we identify {−1, 1} with {−1 mod N, 1 mod N } ⊂ Z N .

Periodic tiling conjecture in one dimension
In this section we adapt the pigeonholing argument of Newman [N77] to establish Theorem 2.1 (One-dimensional case of periodic tiling conjecture).Let G = Z × G 0 for a some explicit finite abelian group G 0 , let J ⩾ 1 be a standard integer, let F 1 , . . ., F J be finite subsets of G, and let E be a periodic subset of G. Then the tiling equation Tile(F 1 , . . ., F J ; E) is not aperiodic (and hence also decidable).
We remark that the same argument also applies to systems of tiling equations in one-dimensional groups Z × G 0 ; this also follows from the above theorem and Theorem 1.15.
We abbreviate the "thickened interval" {n ∈ Z : a ⩽ n ⩽ b} × G 0 as [[a, b]] for any integers a ⩽ b.Since the F 1 , . . ., F J are finite, there exists a natural number L such that F 1 , . . ., F J ⊂ [[−L, L]].Since E is periodic, there exists a natural number r such that E + (n, 0) = E for all n ∈ rZ, where we view (n, 0) as an element of Z × G 0 .We can assign each n ∈ rZ a "color", defined as the tuple . This is a tuple of J subsets of the finite set [[−L, L]], and thus there are only finitely many possible colors.By the pigeonhole principle, one can thus find a pair of integers n 0 , n 0 + D ∈ rZ with D > L that have the same color, thus or equivalently for j = 1, . . ., J.
We now define the sets A ′ j for j = 1, . . ., J by taking the portion A j ∩[[n 0 , n 0 + D − 1]] of A j and extending periodically by DZ × {0}, thus

Clearly we have the agreement
and similarly and thus A j , A ′ j in fact agree on a larger region: (2. 2) It will now suffice to show that A ′ 1 , . . ., A ′ J solve the tiling equation Tile(F 1 , . . ., F J ; E), that is to say that Since both sides of this equation are periodic with respect to translations by DZ × {0}, it suffices to establish this claim within [[n 0 , n 0 + D − 1]], that is to say 2) we may replace each A ′ j in (2.3) by A j .Since A 1 , . . ., A J solve the tiling equation Tile(F 1 , . . ., F J ; E), the claim follows.□ Remark 2.2.An inspection of the argument reveals that the hypothesis that G 0 was abelian was not used anywhere in the proof, thus Theorem 2.1 is also valid for nonabelian G 0 (with suitable extensions to the notation).This generalization will be used in Section 11.

Combining multiple tiling equations into a single equation
In this section we establish Theorem 1.15.For the rest of the section we use the notation and hypotheses of that theorem.
Remark 3.1.The reader may wish to first consider the special case M = 2, J = 1, N = 3 in what follows to simplify the notation.In this case, part (ii) of the theorem asserts that the system of tiling equations We begin with part (ii).Suppose we have a solution ) × {m} for any j = 1, . . ., J and m = 1, . . ., M , and hence by (3.1) But by (1.6), the right-hand side here is Z d × Ẽ0 .Thus we see that Ã1 , . . ., ÃJ solve the single tiling equation Tile( F1 , . . ., FJ ; Conversely, suppose that we have a solution ( Ã1 , . . ., ÃJ ) ∈ Tile( F1 , . . ., FJ ; We claim that Ãj ⊂ G × {0} for all j = 1, . . ., J.For if this were not the case, then there would exist j = 1, . . ., J and an element (g, n) of Ãj with n ∈ Z N \{0}.On the other hand, for any 1 ⩽ m ⩽ M , the set F But since N > M , this is inconsistent with n being a non-zero element of Z N .Thus we have Ãj ⊂ G × {0} as desired, and we may write for some A j ⊂ G.By considering the intersection (or "slice") of (3.2) with G × {m}, we see that for all m = 1, . . ., M , that is to say A 1 , . . ., A J solves the system of tiling equations Tile(F 0 ), m = 1, . . ., M .We have thus demonstrated that the equation Tile( F1 , . . ., FJ ; Z d × Ẽ0 ) admits a solution if and only if the system Tile(F 0 ), m = 1, . . ., M does.This argument is also valid in any other universe U * of ZFC, which gives (ii).An inspection of the argument also reveals that the equation Tile( F1 , . . ., FJ ; Z d × Ẽ0 ) admits a periodic solution if and only if the system Tile(F As noted in the introduction, in view of Theorem 1.15 we see that to prove Theorem 1.8 it suffices to prove Theorem 1.16.This is the objective of the next five sections of the paper. Remark 3.2.For future reference we remark that the abelian nature of G 0 was not used in the above argument, thus Theorem 1.15 is also valid for nonabelian G 0 (with suitable extensions to the notation).

From tiling to functions
In this section we reduce Theorem 1.16 to Theorem 1.17, by means of the following general proposition.
Proposition 4.1 (Equivalence of tiling equations and functional equations).Let G be an explicit finitely generated abelian group, let G 1 be an explicit finite abelian group, let J, M ⩾ 1 and N > J be standard natural numbers, and suppose that for each j = 1, . . ., J and m = 1, . . ., M one is given a (possibly empty) finite subset H (m) j of G and a (possibly empty) subset F (m) j of G 1 .For each m = 1, . . ., M , assume also that we are given a subset E (m) of G 1 .We adopt the abbreviations for integers a ⩽ b.Then the following are equivalent: (i) The system of tiling equations for all m = 1, . . ., M , together with the tiling equations for every permutation σ : {1, . . ., J} → {1, . . ., J} of {1, . . ., J}, admit a solution.(ii) There exist f j : G → G 1 for j = 1, . . ., J that obey the system of functional equations for all n ∈ G and m = 1, . . ., M .
Remark 4.2.The reason why we work with {0} × ) is in order to ensure that one is working with a non-empty tile (as is required in Theorem 1.16), even when the original tile F (m) j is empty.
Remark 4.3.The reader may wish to first consider the special case M = J = 1, N = 2 in what follows to simplify the notation.In this case, the theorem asserts that for any finite H ⊂ G, and F, E ⊂ G 1 , the system of tiling equations for all n ∈ G.The relationship between the set A and the function f will be given by the graphing relation Proof.Let us first show that (ii) implies (i).If f 1 , . . ., f J obey the system (4.3),we define the sets A 1 , . . ., A J ⊂ G × Z N × G 1 to be the graphs of f 1 , . . ., f J in the sense that For any j = 1, . . ., J and permutation σ : {1, . . ., J} → {1, . . ., J}, we have which gives the tiling equation (4.2) for any permutation σ.Next, for j = 1, . . ., J and m = 1, . . ., M , we have and (as a special case of (4.5)) so that the tiling equation (4.1) then follows from (4.3).This shows that (ii) implies (i).Now assume conversely that (i) holds, thus we have sets A 1 , . . ., A J ⊂ G × Z N × G 1 obeying the system of tiling equations for all m = 1, . . ., M , and for all permutations σ : {1, . . ., J} → {1, . . ., J}.

But this is inconsistent with n being a non-zero element of Z
If one considers the intersection (or "slice") of (4.8) with G × [[σ(j)]], we conclude that for any j = 1, . . ., J and permutation σ.This implies that for each n ∈ G there is a unique f j (n) ∈ G 1 such that (n, 0, f j (n)) ∈ A j , thus the A j are of the form (4.4) for some functions f j .The identity (4.6) then holds, and so from inspecting the G×[[0]] "slice" of (4.7) we obtain the equation (4.3).This shows that (ii) implies (i).□ The proof of Proposition 4.1 is valid in every universe U * of ZFC, thus the solvability question in Proposition 4.1(i) is decidable if and only if the solvability question in Proposition 4.1(ii) is.Applying this fact for J = 2, we see that Theorem 1.17 implies Theorem 1.16.
It now remains to establish Theorem 1.17.This is the objective of the next four sections of the paper.

Reduction to the Hamming cube
In this section we show how Theorem 1.18 implies Theorem 1.17.Let N , D, M , h , E (m) be as in Theorem 1.18.For each d = 1, . . ., D, let π d : Z D N → Z N denote the d th coordinate projection, thus y = (π 1 (y), . . ., π D (y)) (5.1) for all y ∈ Z D N .
We write elements of Z 2 × Z 2 as (n, t) with n ∈ Z 2 and t ∈ Z 2 .For a pair of functions f1 , f2 : Z 2 × Z 2 → Z D N , consider the system of functional equations for (n, t) ∈ Z 2 × Z 2 , d = 1, . . ., D and j = 1, 2, as well as the equations for (n, t) ∈ Z 2 × Z 2 and m = 1, . . ., M .Note that this system is of the form (1.7) (with f j replaced by fj , and for suitable choices of M , , E (m) ).It will therefore suffice to establish (using an argument formalizable in ZFC) the equivalence of the following two claims: (i) There exist functions f1 , f2 : Z 2 × Z 2 → Z D N solving the system (5.2),(5.3).(ii) There exist f 1 , f 2 : Z 2 → {−1, 1} D solving the system (1.8).
Remark 5.1.As a simplified version of this equivalence, the reader may wish to take M = 1, D = 2, and only work with a single function f (or f ) instead of a pair f 1 , f 2 (or f1 , f2 ) of functions.The claim is then that the following two statements are equivalent for any F, E ⊂ Z 2 N : (i') There exists f : N obeying the equations: The relation between (i') and (ii') shall basically arise from the ansatz f (n, t) = (−1) t f (n).
It now remains to establish Theorem 1.18.This is the objective of the next three sections of the paper.

Reduction to systems of linear equations on boolean functions
In this section we show how Theorem 1.19 implies Theorem 1.18.Let D, D 0 , M 1 , M 2 , a (m) j,d , h d be as in Theorem 1.19.We let N be a sufficiently large integer.For each j = 1, 2 and m = 1, . . ., M j , we consider the subgroup H . ., D be the coordinate projections as in the previous section.For some unknown functions f 1 , f 2 : N we consider the system of functional equations for all n ∈ Z 2 , j = 1, 2, m = 1, . . ., M j , and ϵ = ±1, as well as the system for all n ∈ Z 2 , d = 1, . . ., D 0 and ϵ = ±1.Note that this system (6.2),(6.3) is of the form required for Theorem 1.18.It will suffice to establish (using an argument valid in every universe of ZFC) the equivalence of the following two claims: (i) There exist functions f 1 , f 2 : Z 2 → Z D N solving the system (6.2),(6.3).(ii) There exist functions f j,d : Z 2 → {−1, 1} solving the system (1.9), (1.10).
Remark 6.1.To understand this equivalence, the reader may wish to begin by verifying two simplified special cases of this equivalence.Firstly, the two (trivially true) statements (i') There exist a function f : Z 2 → {−1, 1} 2 solving the equation (ii') There exist functions f 1 , f 2 : Z 2 → {−1, 1} solving the equation can be easily seen to be equivalent after making the substitution Secondly, for any h ∈ Z 2 , the two (trivially true) statements (i") There exist a functions f 1 , f 2 : Z 2 → {−1, 1} solving the equation for all n ∈ Z 2 and ϵ = ±1.(ii") There exist functions f 1 , f 2 : Z 2 → {−1, 1} solving the equation are also easily seen to be equivalent (the solution sets (f 1 , f 2 ) for (i") and (ii") are identical).
It now remains to establish Theorem 1.19.This is the objective of the next two sections of the paper.
This is already quite close to Theorem 1.19, except that the linear constraints (1.9) have been replaced by antipode-avoiding constraints (7.5).To conclude the proof of Theorem 1.19, we will show that each antipode-avoiding constraint (7.5) can be encoded as a linear constraint of the form (1.9) after adding some more functions.
To simplify the notation we will assume that M 1 = M 2 = M , which one can assume without loss of generality by repeating the vectors ϵ (m) j as necessary.
The key observation is the following.If ϵ = (ϵ 1 , . . ., ϵ D 0 ) ∈ {−1, 1} D 0 and y 1 , . . ., y D 0 ∈ {−1, 1} D 0 , then the following claims are equivalent: Indeed, it is easy to see from the triangle inequality and parity considerations (and the hypothesis D 0 ⩾ 2) that (a) and (b) are equivalent, and that (b) and (c) are equivalent.The point is that the antipode-avoiding constraint (a) has been converted into a linear constraint (c) via the addition of some additional variables.
We now set D := D 0 + M (D 0 − 2) and consider the question of whether there exist functions f j,d : Z 2 → {−1, 1}, for j = 1, 2, d = 1, . . ., D that solve the linear system of equations , as well as the linear system (7.1) for j = 1, 2, n ∈ Z 2 , and d = 1, . . ., D 0 .In view of the equivalence of (a) and (c) (and the fact that for each j = 1, 2, m = 1, . . ., M , and n ∈ Z 2 , the variables f j,D 0 +(m−1)(D 0 −2)+d (n) appear in precisely one constraint, namely the equation (7.7) for the indicated values of j, m, n) we see that this system of equations (7.6), (7.1) admits a solution if and only if the system of equations (7.5), (7.6) admits a solution.This argument is valid in every universe of ZFC, hence the solvability of the system (7.6), (7.1) is undecidable.This completes the derivation of Theorem 1.19 from Theorem 1.20.
It now remains to establish Theorem 1.20.This is the objective of the next section of the paper.

Undecidability of local Boolean constraints
In this section we prove Theorem 1.20, which by the preceding reductions also establishes Theorem 1.8.
Our starting point is the existence of an undecidable tiling equation for some standard J and some finite F 1 , . . ., F J ⊂ Z 2 .This was first shown11 in [B66] (after applying the reduction in [G70]), with many subsequent proofs; see for instance [JV20] for a survey.One can for instance take the tile set in [O09], which has J = 11, though the exact value of J will not be of importance here.
Note that to any solution (A 1 , . . ., A J ) ∈ Tile(F 1 , . . ., F J ; Z 2 ) U in Z 2 of the tiling equation ( 8.1), one can associate a coloring function c : Z 2 → C taking values in the finite set {j} × F j by defining c(a j + h j ) := (j, h j ) whenever j = 1, . . ., J, a j ∈ A j , and h j ∈ F j .The tiling equation (8.1) ensures that the coloring function c is well-defined.Furthermore, from construction we see that c obeys the constraint for all n ∈ Z 2 , j = 1, . . ., J, and h j , h ′ j ∈ F j .Conversely, suppose that c : Z 2 → C is a function obeying (8.2).Then if we define A j for each j = 1, . . ., J to be the set of those a j ∈ Z 2 such that c(a j + h j ) = (j, h j ) for some h j ∈ F j , from (8.2) we have c(a j + f ′ j ) = (j, f ′ j ) for all j = 1, . . ., J, a j ∈ A j , and f ′ j ∈ F j , which implies that A 1 , . . ., A J solve the tiling equation (8.1).Thus the solvability of (8.1) is equivalent to the solvability of the equation (8.2); as the former is undecidable in ZFC, the latter is also, since the above arguments are valid in every universe of ZFC.
Since the set C = J j=1 {j} × F j is finite, one can establish an explicit bijection ι : C → Ω between this set and some subset Ω of {−1, 1} D for some D. Composing c with this bijection, we see that the question of locating Boolean functions f 1 , . . ., f D : Z 2 → {−1, 1} obeying the constraints for all n ∈ Z 2 , j = 1, . . ., J, and h j , h ′ j ∈ F j , is undecidable in ZFC.However, this set of constraints is of the type considered in Theorem 1.20 (after enumerating the set of differences {h j − h ′ j : j = 1, . . ., J; h j , h ′ j ∈ F j } as h 1 , . . ., h L for some L, and combining the various constraints in (8.3), (8.4)), and the claim follows.9. Proof of Theorem 1.9 In this section we modify the ingredients of the proof of Theorem 1.8 to establish Theorem 1.9.The proofs of both theorems proceed along similar lines, and in fact are both deduced from a common result in Theorem 1.18; see Figure 1.1.
We begin by proving the following analogue of Theorem 1.15.
Proof.We will just prove (i); the proof of (ii) is similar and is left to the reader.The argument will be a "pullback" of the corresponding proof of Theorem 1.15(i).First, suppose that the system Tile(F J ; E (m) ), m = 1, . . ., M of tiling equations has a periodic solution A 1 , . . ., A J ⊂ Z d , thus Thus we have a periodic solution for the system Tile( F1 , . . ., FJ ; Ẽ).
Theorem 9.2 (An undecidable system of tiling equations with two tiles in Z d ).There exist standard natural numbers d, M , and for each m = 1, . . ., M there exist finite non-empty sets F ; E (m) ), m = 1, . . ., M is undecidable.
We will show that Theorem 1.18 implies Theorem 9.2.In order for the arguments from Section 4 to be effectively pulled back, we will first need to construct a rigid tile that can encode a finite group Z k /Λ as the solution set to a tiling equation.
Lemma 9.3 (A rigid tile).Let N 1 , . . ., N k ⩾ 5, and let Λ ⩽ Z k be the lattice Then there exists a finite subset R of Z k with the property that the solution set Tile(R; Z k ) U of the tiling equation Tile(R; Z k ) consists precisely of the cosets h + Λ of Λ, that is to say Proof.As a first guess, one could take R to be the rectangle which gives (9.7) upon applying π.This completes the derivation of (ii) from (i). □ The proof of Proposition 9.4 is valid in every universe U * of ZFC, so in particular the problem in Proposition 9.4(i) is undecidable if and only if the one in Proposition 9.4(ii) is.Hence, to prove Theorem 9.2, it will suffice to establish the following analogue of Theorem 1.17, in which Z 2 × Z 2 is pulled back to Z 2 × Z.
Theorem 9.5 (An undecidable system of functional equations in Z 2 × Z).
There exists an explicit finite abelian group G 0 , a standard integer M ⩾ 1, and for each m = 1, . . ., M there exist (possibly empty) finite subsets H , E (m) ⊂ G 0 for m = 1, . . ., M such that the question of whether there exist functions g 1 , g 2 : Z 2 × Z → G 0 that solve the system of functional equations for all n ∈ Z 2 × Z and m = 1, . . ., M is undecidable (when expressed as a first-order sentence in ZFC).
We can now prove this theorem, and hence Theorem 1.9, using Theorem 1.18: Proof.We repeat the arguments from Section 5. Let N , D, M , h , E (m) be as in Theorem 1.18.We recall the systems (5.2), (5.3) of functional equations, introduced in Section 5.
As before, for each d = 1, . . ., D, let π d : Z D N → Z N denote the d th coordinate projection.We write elements of Z 2 × Z 2 as (n, t) with n ∈ Z 2 and t ∈ Z 2 and elements of Z 2 × Z as (n, z) with n ∈ Z 2 and z ∈ Z.
For a pair of functions g 1 , g 2 : Z 2 × Z → Z D N , consider the system of functional equations for d = 1, . . ., D and j = 1, 2, as well as the equations for m = 1, . . ., M .It will suffice to establish (using an argument valid in every universe of ZFC) the equivalence of the following two claims: (i) There exist functions f1 , f2 : Z 2 × Z 2 → Z D N solving the systems (5.2), (5.3).(ii) There exist functions g 1 , g 2 : Z 2 × Z → Z D N solving the systems (9.9), (9.10).Indeed, if (i) is equivalent to (ii), by Section 5, (ii) is equivalent to the existence of functions f 1 , f 2 : Z 2 → {−1, 1} D solving the system (1.8).Hence Theorem 1.18 implies Theorem 9.5.

Single tile versus multiple tiles
In this section we continue the comparison between tiling equations for a single tile J = 1, and for multiple tiles J > 1.In the introduction we have already mentioned the "dilation lemma" [B20, Proposition 3.1], [GT20, Lemma 3.1], [T95] that is a feature of tilings of a single tile F that has no analogue for tilings of multiple tiles F 1 , . . ., F J .Another distinction can be seen by taking the Fourier transform.For simplicity let us consider a tiling equation of the form Tile(F 1 , . . ., F J ; Z D ).In terms of convolutions, this equation can be written as Taking distributional Fourier transforms, one obtains (formally, at least) where δ is the Dirac distribution.When J > 1, this equation reveals little about the support properties of the distributions 1 A j .But when J = 1, the above equation becomes 1 A 1 F = δ which now provides significant structural information about the Fourier transform of 1 A ; in particular this Fourier transform is supported in the union of {0} and the zero set of 1 F (which is a trigonometric polynomial).Such information is consistent with the known structural theorems about tiling sets arising from a single tile; see e.g., [GT20,Remark 1.8].Such a rich structural theory does not seem to be present when J ⩾ 2. Now we present a further structural property of tilings of one tile that is not present for tilings of two or more tiles, which we call a "swapping property".We will only state and prove this property for one-dimensional tilings, but it is conceivable that analogues of this result exist in higher dimensions.
Theorem 10.1 (Swapping property).Let G 0 be a finite abelian group, and for any integers a, b we write 1) be subsets of Z × G 0 which agree on the left in the sense that whenever n ⩽ −n 0 for some n 0 .Suppose also that there is a finite subset Then we also have A (ω) ⊕ F = A (0) ⊕ F for any function ω : Z → {0, 1}, where is a subset of Z × G 0 formed by mixing together the fibers of A (0) and A (1) .
Proof.For any n ∈ Z and j = 0, 1, we define the slices A By inspecting the intersection (or "slice") of ( 10.1) at [[n]] for some integer n, we see that (Note that all but finitely many of the terms in these disjoint unions are empty.)In terms of convolutions on the finite abelian group G 0 , this becomes To analyze this equation we perform Fourier analysis on the finite abelian group G 0 .Let G 0 be the Pontryagin dual of G 0 , that is to say the group of homomorphisms ξ : Applying this Fourier transform to (10.3), we conclude that l∈Z f n−l (ξ) 1 F l (ξ) = 0 (10.4) for all n ∈ Z and ξ ∈ G 0 .
Suppose ξ ∈ G 0 is such that 1 F l (ξ) is non-zero for at least one integer l.Let l ξ be the smallest integer with 1 F l ξ (ξ) ̸ = 0, then we can rearrange (10.4) as for all integers n.Since f n (ξ) vanishes for all n ⩽ n 0 , we conclude from induction that f n (ξ) in fact vanishes for all n.
To summarize so far, for any ξ ∈ G 0 , either 1 F l (ξ) vanishes for all l, or else f n (ξ) vanishes for all n.In either case, we see that we can generalize (10.4) to l∈Z ω(n − l) f n−l (ξ) 1 F l (ξ) = 0 for all n ∈ Z and ξ ∈ G 0 .Inverting the Fourier transform, this is equivalent to for j = 0, 1, where a (0) , a (1) : Z → Z 2 are two functions that agree at negative integers.Then we have A (0) ⊕ F = A (1) ⊕ F = Z × G 0 .Furthermore, for any ω : Z → {0, 1}, the set satisfies the same tiling equation: Example 10.3.Let G 0 = Z 2 , F = {(0, 0), (1, 1)}, and let A (j) := {(n, j) : n ∈ Z} for j = 0, 1.Then, as in the previous example, we have will not obey the same tiling equation: The problem here is that A (0) , A (1) do not agree to the left.Thus we see that this hypothesis is necessary for the theorem to hold.
Informally, Theorem 10.1 asserts that if E ⊂ Z × G 0 for a finite abelian group G 0 and F is a finite subset of Z × G 0 , then the solution space Tile(F ; E) U to the tiling equation Tile(F ; E) has the following "swapping property": any two solutions in this space that agree on one side can interchange their fibers arbitrarily and remain in the space.This is quite a strong property that is not shared by many other types of equations.Consider for instance the simple equation constraining two Boolean functions f 1 , f 2 : Z → {−1, 1}; this is a specific case of the equation (1.10).We observe that this equation does not obey the swapping property.Indeed, consider the two solutions (f 2 ), (f 2 ) to (10.5) given the formula f (i) j (n) = (−1) 1 n>i+j for i = 0, 1 and j = 1, 2. These two solutions agree on the left, but for a given function ω : Z → {0, 1}, the swapped functions only obeys (10.5) when ω(1) = ω(2).Because of this, unless the equations (10.5) are either trivial or do not admit any two different solutions that agree on one side, it does not seem possible to encode individual constraints such as (10.5) inside tiling equations Tile(F ; E) involving a single tile F , at least in one dimension.As such constraints are an important component of our arguments, it does not seem particularly easy to adapt our methods to construct undecidable or aperiodic tiling equations for a single tile.We remark that in the very special case of deterministic tiling equations, such as the aperiodic tiling equations that encode the construction of Kari in [K96], this obstruction is not present, for then if two solutions to (10.5) agree on one side, they must agree everywhere13 .So it may still be possible to encode such equations inside tiling equations that consist of one tile.
However, as was shown in the previous sections, we can encode any system of equations of the type (10.5) in a system of tiling equations involving more than one tile.
The obstruction provided by Theorem 10.1 relies crucially on the abelian nature of G 0 (in order to utilize the Fourier transform), suggesting that this obstruction is not present in the nonabelian setting.This suggestion is validated by the results in Section 11 below.for all ℓ = 1, . . ., L. Applying (11.11)  for all d = 1, . . ., D.
Next, from (11.14) we have in particular that In the one dimensional case, the two formulations are equivalent (see [LW96]).In the two dimensional case the precise relationship between the discrete and continuous formulations of the periodic tiling conjecture is not known.In [Ken92,Ken93] Kenyon extended the result in [G-BN91] and proved that the periodic tiling conjecture holds for topological discs in R 2 .In [GT20] we proved that for any finite F ⊂ Z 2 and periodic E ⊂ Z 2 , all the solutions to the equation Tile(F, E) are weakly periodic.This implies a similar result for some special types of tile F in R 2 , by using the construction in Remark 1.10.We hope to extend this class of tiles and consider the higher dimensional case of Problem 12.2 in a future work.12.3.We suggest several possible improvements of our construction.
• It might be possible to modify our argument to allow E 0 in Theorem 1.8 to equal G 0 .
Problem 12.3.Is there any finite abelian group G 0 for which there exist finite non-empty sets F 1 , F 2 ⊂ Z 2 × G 0 such that the tiling equation Tile(F 1 , F 2 ; Z 2 × G 0 ) is undecidable?
• In [G-S99] a construction of two tiles F 1 , F 2 in R 2 is given in which the tiling equation is aperiodic if one is allowed to apply arbitrary isometries (not just translations) to the tiles F 1 , F 2 ; each tile ends up lying in eight translation classes, so in our notation this is actually an aperiodic construction with J = 2 × 8 = 16.Similarly for the "Ammann A2" construction in [AGS92] (with J = 2 × 4 = 8).The aperiodic tiling of R 2 (or the hexagonal lattice) construction in [ST11] involves a class of twelve tiles that are all isometric to a single tile (twelve being the order of the symmetry group of the hexagon).
It may be possible to adapt the construction used to prove Theorem 1.8 so that the tiles F 1 , F 2 are isometric to each other.On the other hand, we note a remarkable result of Gruslys,Leader,and Tan [GLT16] that asserts that for any non-empty finite subset F of Z d , there exists a tiling of Z n for some n ⩾ d by isometric copies of F .Problem 12.4.Does our construction provide an example of a finite abelian group G 0 , a subset E 0 ⊂ G 0 , and two finite sets F 1 , F 2 ⊂ Z 2 × G 0 which are isometric to each other, such that the tiling equation is undecidable?
• The finite abelian group G 0 in Theorem 1.8 obtained from our construction is quite large.It would be interesting to optimize the size of G 0 .
Problem 12.5.Find the smallest finite abelian group G 0 for which there exist finite non-empty sets F 1 , F 2 ⊂ Z 2 × G 0 , and E 0 ⊂ G 0 such that the tiling equation Tile(F 1 , F 2 ; Z 2 × E 0 ) is undecidable.
• It might be possible to reduce the dimension d in Theorem 1.9 by "folding" more efficiently the finite construction of G 0 in Theorem 1.8, into a lower dimensional infinite space.
π d : Z D N → Z N for d = 1, .

Figure 9
Figure 9.1.A tiling by the rigid tile R constructed in Lemma 9.3.

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Z d and periodic sets E (m) ⊂ Z d such that the system of tiling equations Tile(F for all n ∈ Z and x ∈ G 0 .If one now introduces the functions f n : G 0 → C for n ∈ Z by the formulaf n := 1 A (1) n − 1 A (0)n then by hypothesis f n vanishes for n ⩽ n 0 , and alsol∈Z f n−l * 1 F l (x) = 0 (10.3)forevery n ∈ Z and x ∈ G.