The Coin Exchange Problem and the Structure of Cube Tilings
Keyword(s):
It is shown that if $[0,1)^d+t$, $t\in T$, is a unit cube tiling of $\mathbb{R}^d$, then for every $x\in T$, $y\in \mathbb{R}^d$, and every positive integer $m$ the number $|T\cap (x+\mathbb{Z}^d)\cap([0,m)^d+ y)|$ is divisible by $m$. Furthermore, by a result of Coppersmith and Steinberger on cyclotomic arrays it is proven that for every finite discrete box $D=D_1\times\cdots\times D_d \subseteq x+\mathbb{Z}^d$ of size $m_1\times \cdots\times m_d$ the number $|D\cap T|$ is a linear combination of $m_1,\ldots, m_d$ with non-negative integer coefficients. Several consequences are collected. A generalization is presented.
2015 ◽
Vol 11
(08)
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pp. 2505-2511
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Keyword(s):
2015 ◽
Vol DMTCS Proceedings, 27th...
(Proceedings)
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1954 ◽
Vol 6
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pp. 449-454
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1971 ◽
Vol 3
(02)
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pp. 200-201
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2011 ◽
Vol 07
(06)
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pp. 1603-1614
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