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- W1796555025 abstract "A cube tiling of $mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T={[0,1)^d+t:tin T}$ such that $bigcup_{tin T}([0,1)^d+t)=mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $jin [d]={1,ldots, d}$ and $t_i=s_i$ for every $iin [d]setminus {j}$. 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 $dleq 6$ and false for all dimensions $dgeq 8$. For $d=7$ the conjecture is still open. Let $xin mathbb{R}^d$, $iin [d]$, and let $L(T,x,i)$ be the set of all $i$th coordinates $t_i$ of vectors $tin T$ such that $([0,1)^d+t)cap ([0,1]^d+x)neq emptyset$ and $t_ileq x_i$. It is known that if $|L(T,x,i)|leq 2$ for some $xin mathbb{R}^7$ and every $iin [7]$ or $|L(T,x,i)|geq 6$ for some $xin mathbb{R}^7$ and $iin [7]$, 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 $xin mathbb{R}^7$ and $iin [7]$. Thus, if there is a counterexample to Keller's conjecture in dimension seven, then $|L(T,x,i)|in {3,4}$ for some $xin mathbb{R}^7$ and $iin [7]$." @default.
- W1796555025 created "2016-06-24" @default.
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- W1796555025 date "2014-01-19" @default.
- W1796555025 modified "2023-09-28" @default.
- W1796555025 title "On Keller's conjecture in dimension seven" @default.
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