2014/01/19 by Andrzej P. Kisielewicz, Kisielewicz, Andrzej P., Magdalena Łysakowska +1
Mathematics · #52C22 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #math.CO #math.MG #msc:52C22
paper · pdf · doi:10.48550/arxiv.1401.4689
37 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1304.1639
arxiv created 2014/05/23 · arxiv updated 2014/05/26
A cube tiling of ℝd is a family of pairwise disjoint cubes [0,1)d+T=\[0,1)d+t:t∈ T\ such that \bigcupt∈ T([0,1)d+t)=ℝd. Two cubes [0,1)d+t, [0,1)d+s are called a twin pair if |tj-sj|=1 for some j∈ [d]=\1,…, d\ and ti=si for every i∈ [d]∖ \j\. 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)≠ ∅ and ti≤ xi. It is known that if |L(T,x,i)|≤ 2 for some x∈ ℝ7 and every i∈ [7] or |L(T,x,i)|≥ 6 for some x∈ ℝ7 and i∈ [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∈ ℝ7 and i∈ [7]. Thus, if there is a counterexample to Keller's conjecture in dimension seven, then |L(T,x,i)|∈ \3,4\ for some x∈ ℝ7 and i∈ [7].