# Towards Resolving Keller's Cube Tiling Conjecture in Dimension Seven

## TL;DR

This paper advances the understanding of Keller's cube tiling conjecture in dimension seven by proving it holds when the maximum number of certain coordinate sets is four, narrowing down the last unresolved case.

## Contribution

The paper proves Keller's conjecture in dimension seven for the case where the maximum coordinate set size is four, completing the analysis for this critical case.

## Key findings

- Keller's conjecture holds for dimension 7 when r^+(T)=4.
- A new proof of Keller's conjecture for dimensions up to 6 is provided.
- If a counterexample exists in dimension 7, then r^+(T)=3, the last open case.

## Abstract

A cube tiling of $\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t\colon t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t)=\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\in [d]=\{1,\ldots, d\}$ and $t_i=s_i$ for every $i\in [d]\setminus \{j\}$. In $1930$, Keller conjectured that in every cube tiling of $\mathbb{R}^d$ there is a twin pair. Keller's conjecture is true for dimensions $d\leq 6$ and false for all dimensions $d\geq 8$. For $d=7$ the conjecture is still open. Let $x\in \mathbb{R}^d$, $i\in [d]$, and let $L(T,x,i)$ be the set of all $i$th coordinates $t_i$ of vectors $t\in T$ such that $([0,1)^d+t)\cap ([0,1]^d+x)\neq \emptyset$ and $t_i\leq x_i$. Let $r^-(T)=\min_{x\in \mathbb{R}^d}\; \max_{1\leq i\leq d}|L(T,x,i)|$ and $r^+(T)=\max_{x\in \mathbb{R}^d}\; \max_{1\leq i\leq d}|L(T,x,i)|$. It is known that if $r^-(T)\leq 2$ or $r^+(T)\geq 5$, then Keller's conjecture is true for $d=7$. In the paper we show that it is also true for $d=7$ if $r^+(T)=4$. Thus, if $[0,1)^7+T$ is a counterexample to Keller's conjecture, then $r^+(T)=3$, which is the last unsolved case of Keller's conjecture. Additionally, a new proof of Keller's conjecture in dimensions $d\leq 6$ is given.