Rigid polyboxes and Keller's conjecture

摘要

A cube tiling of ?d is a family of axis-parallel pairwise disjoint cubes [0,1)d + T = {[0,1)d+t : t ∈ T} that cover ?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. In 1930, Keller conjectured that in every cube tiling of ?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 ∈ ?d, 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. Let r?(T) = minx∈?d max1≤i≤d|L(T,x,i)| and r+(T) = maxx∈?d max1≤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}.