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Some variants of the periodic tiling conjecture

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Pith's one-line read Every finite-tile multi-tiling of the square grid has a periodic twin.

desk verdict A strong, significant paper with a repairable gap in the main proof: the §9.4 density claim is false as stated, but the fix appears straightforward. read the letter →

arxiv 2505.06757 v1 pith:NQCAHDZY submitted 2025-05-10 math.CA math.DSmath.LO

Rachel Greenfeld , Terence Tao This is my paper · ORCID
classification math.CAmath.DSmath.LO MSC 52C2303B2537B52
keywords periodictilingconjecturemulti-tilingstranslationaltilingsdecidabilityfinitelygeneratedabeliangroupsstructuretheoremvanishingsumsofrootsunityrationalization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves three variants of the periodic tiling conjecture, each saying that if a tiling-type equation has any solution at all, it also has one that repeats periodically. The headline result, in the plane grid $\mathbb{Z}^2$, allows the right-hand side to be any periodic integer function and keeps the objects on the left as indicator functions: whenever $f * 1_A = g$ has a set solution $A$, it also has a periodic set solution $A_p$. This settles the conjecture for multi-tilings of $\mathbb{Z}^2$ at every level $k$, where $1_F * 1_A = k$. Along the way the paper shows that homogeneous integer convolution equations $f * a = 0$ always have periodic integer solutions when they have any integer solution, and that the solvability of such equations is algorithmically decidable; it also gives the decidability of multi-tilings in $\mathbb{Z}^2$.

What carries the argument

The engine is a structure theorem (Theorem 3.4) for bounded integer solutions on $\mathbb{Z}^2$: any solution $a$ of $f * a = g$ decomposes as $a = \tilde g - \sum_{w \in W} \varphi_w$, where $\tilde g$ is periodic and each $\varphi_w$ is periodic along a distinct primitive direction $w$, together with a slicing lemma saying that each convolution $(1_{x+\langle w\rangle} f) * \varphi_w$ is periodic. This is derived from the dilation lemma (Lemma 3.1), which says the identity $f * a = g$ is stable under dilating $f$, i.e. $(\tau_r f) * a = g$ for $r \equiv 1 \bmod q$. A second, model-theoretic ingredient is the rationalization Proposition 7.6: a system of linear inequalities with rational coefficients that has a real solution has a rational solution, provided a non-degeneracy condition holds; this is what converts the real coefficients appearing in the "equidistributed" parts of a multi-tiling into rational ones, forcing periodicity.

What would settle it

Exhibit one finite subset $F$ of $\mathbb{Z}^2$ and one periodic integer-valued $g$ for which $f * 1_A = g$ has a set solution $A$ but no periodic set solution; equivalently, find a bounded integer solution $a$ of some $f * a = g$ that cannot be decomposed as $\tilde g - \sum \varphi_w$ with the stated one-directional periodicities and periodic slices. Either example would disprove Theorem 1.9 or Theorem 1.8 and is directly checkable by searching over increasing periods.

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Extended reading notes

Core claim

The central claim, stated as Theorem 1.9, is that for any finitely supported integer-valued $f$ on $\mathbb{Z}^2$ and any periodic integer-valued $g$, if $f * 1_A = g$ for some subset $A$ of $\mathbb{Z}^2$, then $f * 1_{A_p} = g$ for some periodic subset $A_p$. In particular, the periodic tiling conjecture is true for level-$k$ multi-tilings $1_F * 1_A = k$ in $\mathbb{Z}^2$ for every natural number $k$. Two supporting discoveries extend the same principle to weaker settings: the homogeneous equation $f * a = 0$ over the integers admits a nonzero periodic solution whenever it admits any nonzero bounded integer solution (Theorem 1.3, with a proof supplied by Tim Austin), and the non-homogeneous equation $f * a = g$ over the integers in $\mathbb{Z}^2$ admits a periodic integer solution whenever it has any bounded integer solution (Theorem 1.8). The paper also derives decidability statements from these results.

Load-bearing premise

The load-bearing premise is the structure theorem: every bounded integer solution on $\mathbb{Z}^2$ of $f * a = g$ with fixed periodic $g$ can be written as a periodic function minus finitely many functions each periodic in one direction, with slice convolutions also periodic; if a single solution escaped this description, the periodic-replacement argument would fail.

Editorial extensions

If this is right

  • There is an algorithm that decides, for any finitely supported $f$ on $\mathbb{Z}^2$ and periodic integer $g$, whether a set $A$ with $f * 1_A = g$ exists (Corollary 1.10).
  • There is an algorithm that decides, for any finitely generated abelian group $G$ and finitely supported integer $f$, whether $f * a = 0$ has a nonzero bounded integer solution (Corollary 1.5).
  • Level-$k$ multi-tilings of $\mathbb{Z}^2$ by a finite tile $F$ always have periodic counterparts, for every natural number $k$.
  • In rank-one groups $\mathbb{Z} \times H$ with $H$ finite, any integer solution of $f * a = g$ with periodic $g$ can be replaced by a periodic one (Proposition 1.7).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because Theorem 1.8 reduces solvability of $f * a = g$ to the existence of a periodic integer solution, the decidability question for integer tilings (Question 1.11) now hinges on bounding the period or sup norm of such a periodic solution; an effective bound would close the problem.
  • The rationalization Proposition 7.6 has a life beyond tilings: any constraint set that is locally a finite boolean combination of rational linear inequalities admits rational witnesses whenever real witnesses exist, so it could serve as a general tool for turning real-parameter aperiodic constructions into periodic ones.
  • If the structural picture from Theorem 3.4 extends to $\mathbb{Z}^d$ for $d \geq 3$ with periodic right-hand sides, the obstruction to periodicity likely remains concentrated in linear parameters; testing $d = 3$ with a non-constant level function would indicate whether the known high-dimensional failure of the indicator-function conjecture is already visible in the integer-valued setting.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper establishes three variants of the periodic tiling conjecture. First, for f*a=0 with f finitely supported and a bounded integer-valued, existence of a nonzero integer solution implies existence of a nonzero periodic integer solution (a result attributed to Tim Austin), together with decidability of this problem via Theorem 1.3, vanishing sums of roots of unity, and Szmielew's decidability theorem. Second, in Z^2, every bounded integer-valued solution to f*a=g with g periodic can be replaced by a periodic integer-valued solution (Theorem 1.8). Third, in Z^2, if f*1_A=g has an indicator-function solution with g periodic integer-valued, then it has a periodic indicator-function solution (Theorem 1.9), in particular settling the periodic tiling conjecture for multi-tilings of Z^2 at constant level. The proof uses the dilation lemma and a structure theorem that decomposes solutions into a periodic part plus finitely many singly periodic parts, followed by a rationalization procedure: real parameters occurring in an equidistributed description are replaced by rational parameters using locally definable linear predicates and a nondegeneracy result (Proposition 7.6).

Significance. If the proofs are completed as intended, this is a major contribution to the periodic tiling conjecture and to the decidability of tiling problems. Theorem 1.9 goes substantially beyond the known k=1 case in Z^2 (Bhattacharya; Greenfeld--Tao) by covering multi-tilings and periodic right-hand sides, despite the existence of non-weakly-periodic higher-level tilings. The paper also gives a clean decidability result for f*a=0 in all finitely generated abelian groups, and for multi-tilings in Z^2. The methods are notable: the systematic use of the dilation lemma and structure theorems, the retraction to Q, and the model-theoretic/local-definability rationalization in Section 7 are elegant and are likely to be reused. The proofs are detailed and the imported tools (Mann's theorem, Szmielew's theorem, Krylov--Bogolyubov, Weyl equidistribution) are cited. The main caveat is the false equidistribution claim in Section 9.4, which is localized and repairable but currently leaves a gap in the proof of Theorem 1.9.

major comments (1)
  1. [§9.4] The proof that the injective map Φ lies in the predicates Λ_x0 and Λ'_{w,b} uses a false density claim. The text states that the orbit {((y1 Φ(e_i) mod 1)_{i=0}^d, (y2 Φ(e_i) mod 1)_{i=0}^d) : y1,y2 ∈ Z} is dense in (R/Z)^{d+1}×(R/Z)^{d+1}, and similarly that {(Φ(e_i) m mod 1)_{i=0}^d : m ∈ Z} is dense in (R/Z)^{d+1}. Since Φ ∈ Hom_1(Q^{d+1},R), we have Φ(e_0)=1, so for integer y1, y2, m the i=0 coordinates are identically 0 mod 1. The correct statement is that the first orbit is dense in {0}×T^d × {0}×T^d and the second in {0}×T^d. Consequently the predicates Λ_x0 and Λ'_{w,b}, as defined with unrestricted r_0,s_0,p'_0, are not shown to contain Φ, and the application of Proposition 7.6 to these predicates is not justified. This is a load-bearing gap in the proof of Theorem 1.9, because the final rationalization step requires a locally definable predicate satisfied by Φ. The repair is direct: drop the e0 coordinate (equivalently fix r_0=s_0=0 and p'_0=0) and work on the d-dimensional torus; then the orbits are dense in T^d×T^d and T^d respectively, and the subsequent local-definability and quantifier-elimination arguments go through. The manuscript should be revised to make this restriction explicit in equations (9.15) and (9.17) and in the definitions of Λ_x0 and Λ'_{w,b}.
minor comments (4)
  1. [Theorem 3.4(i) / Definition 3.3] The wording 'linearly independent primitive elements' is incorrect for Z^2: the proof of Theorem 3.2 only yields pairwise non-collinear primitive directions, and Corollary 8.6(ii) allows three elements such as {e1,e2,e1+e2} in W. The later arguments use only pairwise non-collinearity, so the statement should be corrected to 'pairwise non-collinear' rather than 'linearly independent'.
  2. [§9.4] In the definitions of F_x0 and S'_{w,b,a,x}, the notation \tildeΦ(α_{w,x0,i}) is undefined because α_{w,x0,i} is a scalar and \tildeΦ is defined on Q^{d+1}; these occurrences should be α_{w,x0,i} \tildeΦ(e_i).
  3. [Question 1.12] The question states that the periodic solution a_p should lie in ℓ∞(Z^2,Z)_p, but the context is Z^d; this should be ℓ∞(Z^d,Z)_p.
  4. [§9.4] After the main density repair is made, the sentence identifying the torus with [0,1)^{d+1}×[0,1)^{d+1} should be changed to [0,1)^d×[0,1)^d (and correspondingly in the treatment of (9.17)), to match the reduced coordinates.

Circularity Check

0 steps flagged · score 1.0 of 10

The paper's derivations are self-contained; the only notable reuse of prior work is through citations to the authors' own dilation and structure lemmas, which are proved or adapted with the arguments included inside the paper, so no load-bearing circularity is present.

full rationale

No circular step is exhibited. Lemma 3.1 (dilation lemma) is stated as a routine modification of [B20, GT21, GGRT23] but a proof is supplied in the text. The structure theorem (Theorem 3.2) and its Z^2 refinement (Theorem 3.4) are proved from Lemma 3.1; Theorem 3.4(i) gives a structured-solution decomposition a = g̃ - Σ φ_w, and Theorem 3.4(ii) (slicing lemma) is proved by adapting [GT21, Lemma 5.1] with the argument included. Theorems 1.8 and 1.9 are then obtained by explicit constructions (retractions, periodic replacements, and the rationalization ansatz) rather than by assuming the desired periodic solution. The conditions (9.4) and (9.5) are reduced to (9.6) and (9.12), and the implication from a non-degenerate rational map Φ_Q satisfying these to a periodic indicator solution is demonstrated by summation and Lemma 2.2. The citations to [GT21] and [GGRT23] are real evidence: the cited structural results are not the PTC itself, are proved in the cited works, and are re-proved or adapted here. The reviewer-flagged density gap in §9.4 (the orbit is dense only after dropping the e0 coordinate because Φ(e0)=1) and the statement in Theorem 3.4(i) that W is linearly independent (where the proof of Theorem 3.2 only yields pairwise non-collinear primitive directions) are correctness concerns, not circularity: neither direction assumes the conclusion of Theorem 1.9. Hence the central derivation is not equivalent to its inputs and the paper receives a low score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or invented entities appear; the paper is a pure theorem-proving paper with no fitting. The listed axioms are standard mathematical tools or clearly stated domain presentations. The paper's own dilation lemma and structure theorem are proved internally, though adapted from prior work.

assumptions (7)
  • standard math ZFC with Axiom of Choice (Hahn-Banach, Zorn, generalized limits)
    Used to build retraction homomorphisms ψ:R→Q (Sections 2, 6) and generalized limit functionals flim (Section 2, eq. (2.3)), which define the projection operators π_v.
  • standard math Weyl equidistribution theorem
    Invoked in Lemma 8.5 for moment limits and in Section 9.4 to pass from infinite constraints to locally definable predicates via density of the orbit.
  • standard math Krylov-Bogolyubov theorem
    Used in Proposition 8.3 to obtain an invariant probability measure on the space of structured solutions, making 'bad events' null.
  • standard math Quantifier elimination for linear inequalities over the reals (linear programming / Tarski-Seidenberg)
    Stated in Remark 7.5 and used in Section 9.4 to convert first-order predicates in L to zeroth-order locally definable forms.
  • standard math Mann's theorem on vanishing sums of roots of unity
    Lemma 5.1, the structural control on minimal vanishing sums, used to express P_n in the first-order language of Q/Z.
  • standard math Szmielew's decidability of the theory of divisible Abelian groups
    Theorem 5.4, the decision procedure behind Corollary 1.5.
  • domain assumption Finitely generated Abelian groups are presented as Z^d × ∏ Z/N_m Z
    Section 5 assumes this standard presentation for computability of the decidability algorithm.

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Pith. "Pith review of Some variants of the periodic tiling conjecture." pith.science (2026). https://pith.science/paper/NQCAHDZY

@misc{pith2026250506757,
  author       = {Pith},
  title        = {Pith review of: Some variants of the periodic tiling conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQCAHDZY}},
  note         = {Machine review of arXiv:2505.06757}
}
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 $G=\mathbb{Z}^2$.

Figures

Figures reproduced from arXiv: 2505.06757 by the authors.

Figure 1.1
Figure 1.1. Inclusions between the various Abelian groups of bounded functions on G studied in this paper. We will be interested in solving the equation f ∗ a = g for a given f ∈ ℓ ∞(G, Z)c and g ∈ ℓ ∞(G, Z)p, and some a that will lie in ℓ ∞(G, Z) and will often have additional constraints imposed, such as being periodic or being an indicator function 1A. A motivating problem in this area is the periodic tiling conjecture (PTC)… view at source ↗

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Works this paper leans on

32 extracted references · 31 canonical work pages

  1. [1]

    Austin, Private communication, 2023

    T. Austin, Private communication, 2023

    work page 2023

  2. [2]

    Beauquier, M

    D. Beauquier, M. Nivat, On translating one polyomino to tile the plane, Discrete Comput. Geom. 6 (1991), no. 6, 575--592

    work page 1991

  3. [3]

    Bhattacharya, Periodicity and Decidability of Tilings of ^2 , Amer

    S. Bhattacharya, Periodicity and Decidability of Tilings of ^2 , Amer. J. Math., 142, (2020), 255--266

    work page 2020

  4. [4]

    N. N. Bogoliubov and N. M. Krylov, La theorie generale de la mesure dans son application a l'etude de systemes dynamiques de la mecanique non-lineaire, Annals of Mathematics, Second Series (in French), 38 (1937): 65--113

    work page 1937

  5. [5]

    J. H. Conway, A. J. Jones, Trigonometric Diophantine equations (On vanishing sums of roots of unity), Acta Arith. 30 (1976), no. 3, 229--240

    work page 1976

  6. [6]

    de Dios, J

    J. de Dios, J. Greb\'ik, R. Greenfeld, J. Madrid, Periodicity and decidability of translational tilings by rational polygonal sets, Bent Fuglede memorial volume of Expositiones Mathematicae, 2024

    work page 2024

  7. [7]

    Girault-Beauquier, M

    D. Girault-Beauquier, M. Nivat, Tiling the plane with one tile, Topology and category theory in computer science (Oxford, 1989), Oxford Sci. Publ., Oxford Univ. Press, New York, 1991, 291--333

    work page 1989

  8. [8]

    Greb\'ik, R

    J. Greb\'ik, R. Greenfeld, V. Rozho n , T. Tao, Measurable tilings by Abelian group actions, International Mathematics Research Notices, Volume 2023, Issue 23, December 2023, Pages 20211--20251

    work page 2023

Show all 32 references
  1. [9]

    Greenfeld, M

    R. Greenfeld, M. Kolountzakis, Tiling, spectrality and aperiodicity of connected sets, Israel Journal of Mathematics

  2. [10]

    Greenfeld, T

    R. Greenfeld, T. Tao, The structure of translational tilings in ^d , Discrete Anal. 2021, Paper No. 16, 28 pp

    2021

  3. [11]

    Greenfeld, T

    R. Greenfeld, T. Tao, Undecidable translational tilings with only two tiles, or one non Abelian tile, Discrete & Computational Geometry 70 (4), 1652--1706 (2023)

    2023

  4. [12]

    Greenfeld, T

    R. Greenfeld, T. Tao, A counterexample to the periodic tiling conjecture, Annals of Math., 200, Issue 1 (2024)

    2024

  5. [13]

    Greenfeld, T

    R. Greenfeld, T. Tao, Undecidability of translational monotilings, arXiv:2309.09504

    arXiv

  6. [14]

    J. Kari, M. Szabados, An Algebraic Geometric Approach to Nivat’s Conjecture, In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part II, pages 273--285, 2015

    2015

  7. [15]

    J. Kari, M. Szabados, An algebraic geometric approach to Nivat's conjecture, Information and Computation, 271 (2020), 104481, ISSN 0890-5401

    2020

  8. [16]

    Kenyon, Rigidity of planar tilings, Invent

    R. Kenyon, Rigidity of planar tilings, Invent. Math., 107 (1992), 637--651

    1992

  9. [17]

    Kenyon, Erratum: ``Rigidity of planar tilings'', Invent

    R. Kenyon, Erratum: ``Rigidity of planar tilings'', Invent. Math., 112 (1993), 223

    1993

  10. [18]

    J. C. Lagarias, Y. Wang, Tiling the line with translates of one tile, Invent. Math. 124 (1996), no. 1-3, 341--365

    1996

  11. [19]

    T.Y Lam, K.H Leung, On Vanishing Sums of Roots of Unity, Journal of Algebra 224 1 (2000), 91--109

    2000

  12. [20]

    Leptin, D

    H. Leptin, D. M\" u ller, Uniform partitions of unity on locally compact groups, Adv. Math. 90 (1991), no. 1, 1--14

    1991

  13. [21]

    H. B. Mann, On linear relations between roots of unity, Mathematika 12 (1965) 107--117

    1965

  14. [22]

    McMullen, Convex bodies which tile space by translations, Mathematika 27 (1980), 113--121

    P. McMullen, Convex bodies which tile space by translations, Mathematika 27 (1980), 113--121

    1980

  15. [23]

    Meyerovitch, S

    T. Meyerovitch, S. Sanadhya, Y. Solomon, A note on reduction of tiling problems, Isr. J. Math. (2025)

    2025

  16. [24]

    D. J. Newman, Tesselation of integers, J. Number Theory 9 (1977), no. 1, 107--111

    1977

  17. [25]

    Nivat, Invited talk at the International Colloquium on Automata, Languages and Programming (ICALP), Bologna, 1997

    M. Nivat, Invited talk at the International Colloquium on Automata, Languages and Programming (ICALP), Bologna, 1997

    1997

  18. [26]

    Poonen, M

    B. Poonen, M. O. Rubinstein. The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discret. Math. 11 (1995): 135--156

    1995

  19. [27]

    J. P. Steinberger, Minimal vanishing sums of roots of unity with large coefficients, Proc. Lond. Math. Soc. 97 (2008), no. 3, 689--717

    2008

  20. [28]

    M. Szegedy, Algorithms to tile the infinite grid with finite clusters, Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS ’98), IEEE Computer Society, Los Alamitos, CA, (1998), 137--145

    1998

  21. [29]

    Szmielew, Elementary properties of Abelian groups, Fund Math, 41 (1955), 203--271

    A. Szmielew, Elementary properties of Abelian groups, Fund Math, 41 (1955), 203--271

    1955

  22. [30]

    Tijdeman, Decomposition of the integers as a direct sum of two subsets, Number theory (Paris, 1992--1993), 261--276, London Math

    R. Tijdeman, Decomposition of the integers as a direct sum of two subsets, Number theory (Paris, 1992--1993), 261--276, London Math. Soc. Lecture Note Ser., 215, Cambridge Univ. Press, Cambridge, 1995

    1992

  23. [31]

    B. A. Venkov, On a class of Euclidean polyhedra, Vestnik Leningrad Univ. Ser. Math. Fiz. Him. 9 (1954), 11--31

    1954

  24. [32]

    Wang, Dominoes and the case of the decision problem, Mathematical Theory of Automata pp

    H. Wang, Dominoes and the case of the decision problem, Mathematical Theory of Automata pp. 23--55 (1963)

    1963

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