Keller’s Conjecture on the Existence of Columns in Cube Tilings of R
نویسنده
چکیده
It is shown that if n ≤ 6, then each tiling of R by translates of the unit cube [0, 1) contains a column; that is, a family of the form {[0, 1) + (s + kei) : k ∈ Z}, where s ∈ R n and ei is an element of the standard basis of R.
منابع مشابه
On Keller's Conjecture in Dimension Seven
A cube tiling of R is a family of pairwise disjoint cubes [0, 1) + T = {[0, 1) + t : t ∈ T} such that ⋃ t∈T ([0, 1) d + t) = R. Two cubes [0, 1) + t, [0, 1) + 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 R there is a twin pair. Keller’s conjecture is true for dimensions ...
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