Keller's Conjecture on the Existence of Columns in Cube Tilings of R^n

Magdalena Łysakowska, Krzysztof Przesławski

It is shown that if n<7, then each tiling of R^n by translates of the unit cube [0,1)^n contains a column; that is, a family of the form {[0,1)^n+(s+ke_i): k \in Z}, where s \in R^…