Towards resolving Keller’s cube tiling conjecture in dimension seven

Abstract

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 ∪ t∈T ([0, 1) d + t) = 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 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. For x ∈ ℝ d and i ∈ [d], 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 ) | $r^-(T)=\min_{x\in \mathbb{R}^d} \max_{1\leq i\leq d}|L(T,x,i)|$ and r + ( T ) = max x ∈ R d max 1 ≤ i ≤ d | L ( T , x , i ) | $r^ + (T)=\max_{x\in \mathbb{R}^d} \max_{1\leq i\leq d}|L(T,x,i)|$ . Before 2019 it was known that Keller’s conjecture is true for dimensions d ≤ 6 and false for all dimensions d = 8. Moreover, in dimension 7 it was known to be true if r −(T) ≤ 2 or r +(T) = 5. The present paper resolves the case r +(T) = 4. At the end of 2019, when the paper was still under review, Brakensiek et al. resolved the cases r +(T) ∈ {3, 4, 6}, proving thereby Keller’s conjecture in dimension 7.