The coin exchange problem and the structure of cube tilings

Andrzej Piotr Kisielewicz¹ ; Krzysztof Przeslawski¹

¹Wydział Matematyki, Informatyki i Ekonometrii, Uniwersytet Zielonogórski, Poland

The electronic journal of combinatorics, Tome 19 (2012) no. 2

Résumé

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.

Détail

DOI : 10.37236/2251

Classification : 05B45, 05B40, 11H06, 52C22
Mots-clés : cube tilings