Towards Resolving Keller's Cube Tiling Conjecture in Dimension Seven

A cube tiling of ${\mathbb{R}}^d$ is a family of pairwise disjoint cubes $[0,1{)}^d+T=\{[0,1{)}^d+t\text{\hspace{0.17em}⁣}:t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t)=$\mathbb{R}$^d$\mathrm{.}Twocubes$[0,1)^d+t$,$[0,1)^d+s$ are called a twin pair if $\mathrm{\mid }{t}_j-s_j\mathrm{\mid }=1$ for some $j\in [d]=\{1,\dots ,d\}$ 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 $d\le 6$ and false for all dimensions $d\ge 8$. 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 $([0,1{)}^d+t)\cap ([0,1{]}^d+x)\mathrm{\ne }\mathrm{\varnothing }$ and $t_i\le x_i$. Let $r^{-}(T)={\min }_{x\in {\mathbb{R}}^d}{\max }_{1\le i\le d}\mathrm{\mid }L(T,x,i)\mathrm{\mid }$ and $r^{+}(T)={\max }_{x\in {\mathbb{R}}^d}{\max }_{1\le i\le d}\mathrm{\mid }L(T,x,i)\mathrm{\mid }$. It is known that if $r^{-}(T)\le 2$ or $r^{+}(T)\ge 5$, then Keller’s conjecture is true for $d=7$. In the paper we show that it is also true for $d=7$ if $r^{+}(T)=4$. Thus, if $[0,1{)}^7+T$ is a counterexample to Keller’s conjecture, then $r^{+}(T)=3$, which is the last unsolved case of Keller’s conjecture. Additionally, a new proof of Keller’s conjecture in dimensions $d\le 6$ is given.