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Some variants of the periodic tiling conjecture

Rachel Greenfeld, Terence Tao

TL;DR

The paper develops three variants of the periodic tiling conjecture (PTC) for finitely generated abelian groups, including a dilation-stability framework and a decomposition of solutions to tiling equations. It proves a dilation lemma, a structure theorem, and a slicing lemma that together yield a normal form for solutions to $f*a=g$, enabling both integer-valued and indicator tilings to be analyzed in $\mathbb{Z}^2$, with decidability results for constant-level tilings and multi-tilings. A key advance is the higher-level PTC in $\mathbb{Z}^2$, which establishes the existence of periodic indicator tilings when a tiling exists, by a strategy that cleans the solution, encodes irrational parameters via a rational map, and reduces the problem to a finite set of linear constraints. The work also proves decidability results via connections to vanishing sums of roots of unity and Szmielew’s decidability for divisible abelian groups. Overall, the paper provides a comprehensive framework for transforming real tilings into rational, periodic ones in low dimensions and clarifies the landscape of decidability for tilings in abelian groups.

Abstract

The periodic tiling conjecture (PTC) asserts, for a finitely generated Abelian group $G$ and a finite subset $F$ of $G$, that if there is a set $A$ that solves the tiling equation $\mathbb{1}_F * \mathbb{1}_A = 1$, there is also a periodic solution $\mathbb{1}_{A_{\mathrm{p}}}$. This conjecture is known to hold for some groups $G$ and fail for others. In this paper we establish three variants of the PTC. The first (due to Tim Austin) replaces the constant function $1$ on the right-hand side of the tiling equation by $0$, and the indicator functions $\mathbb{1}_F$ and $\mathbb{1}_A$ by bounded integer-valued functions. The second, which applies in $G=\mathbb{Z}^2$, replaces the right-hand side of the tiling equation by an integer-valued periodic function, and the functions $\mathbb{1}_F$ and $\mathbb{1}_A$ on the left-hand side by bounded integer-valued functions. The third (which is the most difficult to establish) is similar to the second, but retains the property of both $\mathbb{1}_A$ and $\mathbb{1}_{A_{\mathrm{p}}}$ being indicator functions; in particular, we establish the PTC for multi-tilings in $G=\mathbb{Z}^2$. As a result, we obtain the decidability of constant-level integer tilings in any finitely generated Abelian group $G$ and multi-tilings in $G=\mathbb{Z}^2$.

Some variants of the periodic tiling conjecture

TL;DR

The paper develops three variants of the periodic tiling conjecture (PTC) for finitely generated abelian groups, including a dilation-stability framework and a decomposition of solutions to tiling equations. It proves a dilation lemma, a structure theorem, and a slicing lemma that together yield a normal form for solutions to , enabling both integer-valued and indicator tilings to be analyzed in , with decidability results for constant-level tilings and multi-tilings. A key advance is the higher-level PTC in , which establishes the existence of periodic indicator tilings when a tiling exists, by a strategy that cleans the solution, encodes irrational parameters via a rational map, and reduces the problem to a finite set of linear constraints. The work also proves decidability results via connections to vanishing sums of roots of unity and Szmielew’s decidability for divisible abelian groups. Overall, the paper provides a comprehensive framework for transforming real tilings into rational, periodic ones in low dimensions and clarifies the landscape of decidability for tilings in abelian groups.

Abstract

The periodic tiling conjecture (PTC) asserts, for a finitely generated Abelian group and a finite subset of , that if there is a set that solves the tiling equation , there is also a periodic solution . This conjecture is known to hold for some groups and fail for others. In this paper we establish three variants of the PTC. The first (due to Tim Austin) replaces the constant function on the right-hand side of the tiling equation by , and the indicator functions and by bounded integer-valued functions. The second, which applies in , replaces the right-hand side of the tiling equation by an integer-valued periodic function, and the functions and on the left-hand side by bounded integer-valued functions. The third (which is the most difficult to establish) is similar to the second, but retains the property of both and being indicator functions; in particular, we establish the PTC for multi-tilings in . As a result, we obtain the decidability of constant-level integer tilings in any finitely generated Abelian group and multi-tilings in .
Paper Structure (18 sections, 25 theorems, 164 equations, 1 figure)

This paper contains 18 sections, 25 theorems, 164 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Methods of proof
  3. Acknowledgments
  4. Notation
  5. The dilation lemma, and some consequences
  6. Solving f*a=0 in the integers
  7. Solvability of $f*a=0$ in the integers is decidable
  8. Integer-valued PTC in Z^2
  9. Transforming real solutions to rational solutions
  10. Higher level PTC in Z^2, I: cleaning the solution
  11. Strategy
  12. Analyzing the structure of the functions $\varphi_w$
  13. Analyzing the equidistributed case via the moment method
  14. Higher level PTC in Z^2, II: describing the normalized solution
  15. Describing the irrationality of the solution
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $G$ be a finitely generated Abelian group, and let $f \in \ell^\infty(G,\mathbb{C})_{{\operatorname{c}}}$. Then the following are equivalent:

Figures (1)

  • Figure 1.1: Inclusions between the various Abelian groups of bounded functions on $G$ studied in this paper.

Theorems & Definitions (56)

  • Conjecture 1.1: Periodic tiling conjecture
  • Theorem 1.2: Solving $f*a=0$ in the complex numbers
  • proof
  • Theorem 1.3: Solving $f*a=0$ in the integers
  • Example 1.4
  • Corollary 1.5: Solving $f*a=0$ in the integers is decidable
  • Remark 1.6
  • Proposition 1.7: Periodicity of integer tilings in rank one groups
  • Theorem 1.8: Periodicity of integer tilings in $\mathbb{Z}^2$
  • Theorem 1.9: Periodicity of multi-tilings in $\mathbb{Z}^2$
  • ...and 46 more