Geodesic
The structure of translational tilings in $\mathbb{Z}^d$
Discrete analysis (2021) Cet article a éte moissonné depuis la source Scholastica

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We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union of sets that are individually periodic in one direction), but present a counterexample of a higher level tiling of $\mathbb{Z}^2$ that fails to be weakly periodic. We also establish a quantitative version of the two-dimensional periodic tiling conjecture which asserts that any finite tile in $\mathbb{Z}^2$ that admits a tiling, must admit a periodic tiling, by providing a polynomial bound on the period; this also gives an exponential-type bound on the computational complexity of the problem of deciding whether a given finite subset of $\mathbb{Z}^2$ tiles or not. As a byproduct of our structural theory, we also obtain an explicit formula for a universal period for all tilings of a one-dimensional tile.
Publié le : 
@article{DAS_2021_a11,
     author = {Rachel Greenfeld and Terence Tao},
     title = {The structure of translational tilings in $\mathbb{Z}^d$},
     journal = {Discrete analysis},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2021_a11/}
}
TY  - JOUR
AU  - Rachel Greenfeld
AU  - Terence Tao
TI  - The structure of translational tilings in $\mathbb{Z}^d$
JO  - Discrete analysis
PY  - 2021
UR  - http://geodesic.mathdoc.fr/item/DAS_2021_a11/
LA  - en
ID  - DAS_2021_a11
ER  - 
%0 Journal Article
%A Rachel Greenfeld
%A Terence Tao
%T The structure of translational tilings in $\mathbb{Z}^d$
%J Discrete analysis
%D 2021
%U http://geodesic.mathdoc.fr/item/DAS_2021_a11/
%G en
%F DAS_2021_a11
Rachel Greenfeld; Terence Tao. The structure of translational tilings in $\mathbb{Z}^d$. Discrete analysis (2021). http://geodesic.mathdoc.fr/item/DAS_2021_a11/