Rigid polyboxes and keller's conjecture

Abstract

Abstract A cube tiling of ℝ d is a family of axis-parallel pairwise disjoint cubes [0,1) d + T = {[0,1) d +t : t ∈ T} that cover 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. 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 ti of vectors t ∈ T such that ([0,1) d +t) ∩ ([0,1] d +x) ≠ ø and ti ≤ xi . Let r −(T) = min x∈ℝ d max1≤i≤d |L(T,x,i)| and r +(T) = max x∈ℝ 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}.