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}.

On Keller’s Conjecture in Dimension Seven

A cube tiling of $\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t:t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t=\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\in [d]=\{1,\ldots, d\}$ and $t_i=s_i$ for every $i\in [d]\setminus \{j\}$. In $1930$, Keller conjectured that in every cube tiling of $\mathbb{R}^d$ there is a twin pair. Keller's conjecture is true for dimensions $d\leq 6$ and false for all dimensions $d\geq 8$. For $d=7$ the conjecture is still open. Let $x\in \mathbb{R}^d$, $i\in [d]$, and let $L(T,x,i)$ be the set of all $i$th coordinates $t_i$ of vectors $t\in T$ such that $([0,1)^d+t)\cap ([0,1]^d+x)\neq \emptyset$ and $t_i\leq x_i$. Let $r^-(T)=\min_{x\in \mathbb{R}^d}\; \max_{1\leq i\leq d}|L(T,x,i)|$ and $r^+(T)=\max_{x\in \mathbb{R}^d}\; \max_{1\leq i\leq d}|L(T,x,i)|$. It is known that if $r^-(T)\leq 2$ or $r^+(T)\geq 6$, then Keller's conjecture is true for $d=7$. In the present paper we show that it is also true for $d=7$ if $r^+(T)=5$. Thus, if $[0,1)^d+T$ is a counterexample to Keller's conjecture in dimension seven, then $r^-(T),r^+(T)\in \{3,4\}$.

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.