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 $t_i\leq x_i$. It is known that if $|L(T,x,i)|\leq 2$ for some $x\in \mathbb{R}^7$ and every $i\in [7]$ or $|L(T,x,i)|\geq 6$ for some $x\in \mathbb{R}^7$ and $i\in [7]$, then Keller's conjecture is true for $d=7$. In the present paper we show that it is also true for $d=7$ if $|L(T,x,i)|=5$ for some $x\in \mathbb{R}^7$ and $i\in [7]$. Thus, if there is a counterexample to Keller's conjecture in dimension seven, then $|L(T,x,i)|\in \{3,4\}$ for some $x\in \mathbb{R}^7$ and $i\in [7]$.Comment: 37 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1304.163

On the number of neighborly simplices in R^d

Two $d$-dimensional simplices in $R^d$ are neighborly if its intersection is a $(d-1)$-dimensional set. A family of $d$-dimensional simplices in $R^d$ is called neighborly if every two simplices of the family are neighborly. Let $S_d$ be the maximal cardinality of a neighborly family of $d$-dimensional simplices in $R^d$. Based on the structure of some codes $V\subset \{0,1,*\}^n$ it is shown that $\lim_{d\rightarrow \infty}(2^{d+1}-S_d)=\infty$. Moreover, a result on the structure of codes $V\subset \{0,1,*\}^n$ is given