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

TL;DR

This paper proves that for dimensions less than 7, every tiling of R^n with unit cubes necessarily contains a vertical column, advancing understanding of Keller's conjecture in geometric tiling theory.

Contribution

It establishes the existence of columns in cube tilings of R^n for all n<7, providing a significant partial result towards Keller's conjecture.

Findings

For n<7, all cube tilings contain a column.

The result supports Keller's conjecture in lower dimensions.

No such guarantee for n≥7 based on this work.

Abstract

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.