Kellers Conjecture on the Existence of Columns in Cube Tilings of R^n

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^n, e_i is an element of the standard basis of R^n and Z is the set of integers.

💡 Research Summary

The paper addresses a low‑dimensional version of Keller’s conjecture, which asks whether a tiling of ℝⁿ by unit cubes can avoid having two cubes meet only at a vertex. While counter‑examples are known for n ≥ 8, the case n ≤ 6 has remained unresolved. The author introduces the notion of a “column”: an infinite family of cubes aligned along a single coordinate axis, formally {