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Towards resolving Keller’s cube tiling conjecture in dimension seven

  • Andrzej P. Kisielewicz EMAIL logo
From the journal Advances in Geometry

Abstract

A cube tiling of ℝd is a family of pairwise disjoint cubes [0, 1)d + T = {[0, 1)d + t: tT} such that ∪tT([0, 1)d + t) = ℝd. Two cubes [0, 1)d + t, [0, 1)d + s are called a twin pair if |tjsj| = 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. For x ∈ ℝd and i ∈ [d], let L(T, x, i) be the set of all ith coordinates ti of vectors tT such that ([0, 1)d + t)∩([0, 1]d + x)≠∅ and tixi. Let r(T)=minxRdmax1id|L(T,x,i)| and r+(T)=maxxRdmax1id|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.

MSC 2010: 52C22; 05C69; 94B25
  1. Communicated by: M. Joswig

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Received: 2017-07-10
Revised: 2022-01-02
Published Online: 2022-03-04
Published in Print: 2022-04-26

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