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TILING WITH PUNCTURED INTERVALS
Published online by Cambridge University Press: 29 October 2018
Abstract
It was shown by Gruslys, Leader and Tan that any finite subset of $\mathbb{Z}^{n}$ tiles
$\mathbb{Z}^{d}$ for some
$d$. The first non-trivial case is the punctured interval, which consists of the interval
$\{-k,\ldots ,k\}\subset \mathbb{Z}$ with its middle point removed: they showed that this tiles
$\mathbb{Z}^{d}$ 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
$\mathbb{Z}^{4}$.
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- Research Article
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- Copyright
- Copyright © University College London 2018
References
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