Rigid polyboxes and Keller's conjecture

2013/04/05 by Andrzej P. Kisielewicz, Kisielewicz, Andrzej P. · 1 citation
Mathematics · #52C22 #52C25 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #math.CO #math.MG #msc:52C22 #msc:52C25

paper · pdf · doi:10.48550/arxiv.1304.1639

31 pages, 12 figures

arxiv created 2014/12/29 · arxiv updated 2014/12/30

Abstract

A cube tiling of R^d is a family of pairwise disjoint cubes [0,1)^d+T=\[0,1)^d+t:t∈ T\ such that \bigcup_{t∈ T}([0,1)^d+t)=R^d. Two cubes [0,1)^d+t, [0,1)^d+s are called a twin pair if their closures have a complete facet in common, that is if |t_j-s_j|=1 for some j∈ [d]=\1,..., d\ and t_i=s_i for every i∈ [d]∖ \j\. In 1930, Keller conjectured that in every cube tiling of R^d 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∈ R^d, i∈ [d], and let L(T,x,i) be the set of all ith coordinates t_i of vectors t∈ T such that ([0,1)^d+t)∩ ([0,1]^d+x)≠ ∅ and t_i≤ x_i. Let r^-(T)=min_{x∈ R^d} max_{1≤ i≤ d}|L(T,x,i)| and r⁺(T)=max_{x∈ R^d} max_{1≤ 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\.

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• On Keller's conjecture in dimension seven

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