Nivat’s Conjecture Holds for Sums of Two Periodic Configurations

Szabados, Michal

doi:10.1007/978-3-319-73117-9_38

Cited by 7 publications

(16 citation statements)

References 14 publications

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“…The best general bound was proved in [5] where it was shown that for any rectangle D the condition |L D (c)| ≤ |D|/2 is enough to guarantee that c is periodic. This fact can also be proved using the algebraic approach [14].…”

Section: Nivat's Conjecturementioning

confidence: 72%

“…If m = 1 then c is periodic, so it is interesting to consider the cases of m ≥ 2. Szabados proved in [14] that Nivat's conjecture holds in the case m = 2. Note that this case is equivalent to c being the sum of two periodic configurations [9].…”

Section: Contributions To Nivat's Conjecturementioning

confidence: 96%

“…We first proposed this approach in [9,10] to study Nivat's conjecture. It led to a number of subsequent results [6,7,8,14]. In this presentation we review the main results without proofs -the given references can be consulted for more details.…”

Section: Introductionmentioning

confidence: 99%

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Low-Complexity Tilings of the Plane

Kari

2019

Descriptional Complexity of Formal Systems

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A two-dimensional configuration is a coloring of the infinite grid Z 2 with finitely many colors. For a finite subset D of Z 2 , the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The number of distinct D-patterns of a configuration is a natural measure of its complexity. A configuration is considered having low complexity with respect to shape D if the number of distinct D-patterns is at most |D|, the size of the shape. This extended abstract is a short review of an algebraic method to study periodicity of such low complexity configurations.

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Section: Nivat's Conjecturementioning

confidence: 72%

Section: Contributions To Nivat's Conjecturementioning

confidence: 96%

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Low-Complexity Tilings of the Plane

Kari

2019

Descriptional Complexity of Formal Systems

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“…For the second case we apply the technique by Cyr and Kra [8]. This technique was also used in [17] to address Nivat's conjecture. It is possible to use a direct combination of lemmas from [8] or [17] to prove the following:…”

Section: Periodicity In Low Complexity Subshiftsmentioning

confidence: 99%

“…We will prove this proposition below using lemmas from [17]. We first recall some definitions, adjusted to our terminology.…”

Section: Periodicity In Low Complexity Subshiftsmentioning

confidence: 99%

Decidability and Periodicity of Low Complexity Tilings

Kari

Moutot

2021

Theory Comput Syst

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In this paper we study colorings (or tilings) of the two-dimensional grid ${\mathbb {Z}}^{2}$ ℤ 2 . A coloring is said to be valid with respect to a set P of n × m rectangular patterns if all n × m sub-patterns of the coloring are in P. A coloring c is said to be of low complexity with respect to a rectangle if there exist $m,n\in \mathbb {N}$ m , n ∈ ℕ and a set P of n × m rectangular patterns such that c is valid with respect to P and |P|≤ nm. Open since it was stated in 1997, Nivat’s conjecture states that such a coloring is necessarily periodic. If Nivat’s conjecture is true, all valid colorings with respect to P such that |P|≤ mn must be periodic. We prove that there exists at least one periodic coloring among the valid ones. We use this result to investigate the tiling problem, also known as the domino problem, which is well known to be undecidable in its full generality. However, we show that it is decidable in the low-complexity setting. Then, we use our result to show that Nivat’s conjecture holds for uniformly recurrent configurations. These results also extend to other convex shapes in place of the rectangle. After that, we prove that the nm bound is multiplicatively optimal for the decidability of the domino problem, as for all ε > 0 it is undecidable to determine if there exists a valid coloring for a given $m,n\in \mathbb {N}$ m , n ∈ ℕ and set of rectangular patterns P of size n × m such that |P|≤ (1 + ε)nm. We prove a slightly better bound in the case where m = n, as well as constructing aperiodic SFTs of pretty low complexity. This paper is an extended version of a paper published in STACS 2020 (Kari and Moutot 12).

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On periodic decompositions and nonexpansive lines

Colle

2023

Math. Z.

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Nivat’s Conjecture Holds for Sums of Two Periodic Configurations

Cited by 7 publications

References 14 publications

Low-Complexity Tilings of the Plane

Low-Complexity Tilings of the Plane

Decidability and Periodicity of Low Complexity Tilings

On periodic decompositions and nonexpansive lines

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