Rigid polyboxes and keller's conjecture

A cube tiling of ℝd is a family of axis-parallel pairwise disjoint cubes [0,1)d + T = {[0,1)d+t : t ∈ T} that cover ℝd. Two cubes [0,1)d + t, [0,1)d + s are called a twin pair if their closures have a complete facet in common. In 1930, Keller conjectured that in every cube tiling of ℝd there is a twin pair. Keller's conjecture is true for dimensions d ≤ 6 and false for all dimensions d ≥ 8. For d = 7 the conjecture is still open. Let x ∈ ℝd, i ∈ [d], and let L(T, x, i) be the set of all ith coordinates ti of vectors t ∈ T such that ([0,1)d+t) ∩ ([0,1]d+x) ≠ ø and ti ≤ xi. Let r−(T) = minx∈ℝd max1≤i≤d|L(T,x,i)| and r+(T) = maxx∈ℝd max1≤i≤d|L(T,x,i)|. It is known that Keller's conjecture is true in dimension seven for cube tilings [0,1)7 + T for which r−(T) ≤ 2. In the present paper we show that it is also true for d = 7 if r+(T) ≥ 6. Thus, if [0,1)d + T is a counterexample to Keller's conjecture in dimension seven, then r−(T), r+(T) ∈ {3, 4, 5}.

The Coin Exchange Problem and the Structure of Cube Tilings

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.