The coin exchange problem and the structure of cube tilings (Q426903)

Summary: 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 \textit{D. Coppersmith} and \textit{J. Steinberger} [Integers 6, Paper A26, 28 p. (2006; Zbl 1109.11017)] 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.