Tiling with arbitrary tiles

Gruslys, Vytautas; Leader, Imre; Tan, Ta Sheng

doi:10.1112/plms/pdw017

Cited by 14 publications

(40 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Chalcraft [12,13] conjectured that, for any tile T ⊂ Z n , there is some dimension d for which T tiles Z d . This was proved by Gruslys, Leader and Tan [8]. The first non-trivial case is the punctured interval T = XXXXX k .XXXXX k .…”

Section: Introductionmentioning

confidence: 84%

“…We start with an important definition from [8]: a string is a one-dimensional infinite line in Z d with every (k + 1)th point removed. Crucially, a string is a disjoint union of copies of T .…”

Section: Preliminaries and The Odd Casementioning

confidence: 99%

“…For general one-dimensional tiles, Gruslys, Leader and Tan [8] conjectured that there is a bound on the dimension in terms of the size of the tile:…”

Section: Does T Tile Z 3 ?mentioning

confidence: 99%

“…Conjecture 12 (Gruslys, Leader, Tan [8]). For any positive integer t, there exists a number d such that any tile T ⊂ Z with |T | ≤ t tiles Z d .…”

Section: Does T Tile Z 3 ?mentioning

confidence: 99%

“…Question 1 (Gruslys, Leader, Tan [8]). Let T be the punctured interval XXXXX k .XXXXX k , and let d be the least number such that T tiles Z d .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Tiling With Punctured Intervals

Metrebian¹

2018

Mathematika

View full text Add to dashboard Cite

It was shown by Gruslys, Leader and Tan that any finite subset of Z n tiles Z d for some d. The first non-trivial case is the punctured interval, which consists of the interval {−k, . . . , k} ⊂ Z with its middle point removed: they showed that this tiles Z d for d = 2k 2 , and they asked if the dimension needed tends to infinity with k. In this note we answer this question: we show that, perhaps surprisingly, every punctured interval tiles Z 4 .

show abstract

Section: Introductionmentioning

confidence: 84%

Section: Preliminaries and The Odd Casementioning

confidence: 99%

“…For general one-dimensional tiles, Gruslys, Leader and Tan [8] conjectured that there is a bound on the dimension in terms of the size of the tile:…”

Section: Does T Tile Z 3 ?mentioning

confidence: 99%

“…Conjecture 12 (Gruslys, Leader, Tan [8]). For any positive integer t, there exists a number d such that any tile T ⊂ Z with |T | ≤ t tiles Z d .…”

Section: Does T Tile Z 3 ?mentioning

confidence: 99%

“…Question 1 (Gruslys, Leader, Tan [8]). Let T be the punctured interval XXXXX k .XXXXX k , and let d be the least number such that T tiles Z d .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Tiling With Punctured Intervals

Metrebian¹

2018

Mathematika

View full text Add to dashboard Cite

show abstract

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

Greenfeld¹,

Tao²

2023

Discrete Comput Geom

View full text Add to dashboard Cite

We construct an example of a group $$G = \mathbb {Z}^2 \times G_0$$ G = Z 2 × G 0 for a finite abelian group $$G_0$$ G 0 , a subset E of $$G_0$$ G 0 , and two finite subsets $$F_1,F_2$$ F 1 , F 2 of G, such that it is undecidable in ZFC whether $$\mathbb {Z}^2\times E$$ Z 2 × E can be tiled by translations of $$F_1,F_2$$ F 1 , F 2 . In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of E by the tiles $$F_1,F_2$$ F 1 , F 2 , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $$\mathbb {Z}^2$$ Z 2 ). A similar construction also applies for $$G=\mathbb {Z}^d$$ G = Z d for sufficiently large d. If one allows the group $$G_0$$ G 0 to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.

show abstract