Rigid polyboxes and Keller's conjecture

TL;DR

This paper investigates Keller's conjecture on cube tilings in seven dimensions, establishing new conditions under which the conjecture holds, and narrowing down the possible parameters for counterexamples.

Contribution

The paper proves Keller's conjecture in dimension seven for cube tilings with certain bounds on the parameters r^- and r^+, refining the understanding of potential counterexamples.

Findings

Keller's conjecture holds for d=7 when r^+(T) ≥ 6.

Counterexamples in d=7 must have r^-(T), r^+(T) in {3,4,5}.

The result narrows the search for counterexamples to specific parameter ranges.

Abstract

A cube tiling of R^d is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t:t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t)=R^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if their closures have a complete facet in common, that is if $|t_j-s_j|=1$ for some $j\in [d]=\{1,..., d\}$ and $t_i=s_i$ for every $i\in [d]\setminus \{j\}$. In 1930, Keller conjectured that in every cube tiling of 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 $x\in R^d$, $i\in [d]$, 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 $r^-(T)=\min_{x\in R^d}\; \max_{1\leq i\leq d}|L(T,x,i)|$ and $r^+(T)=\max_{x\in R^d}\; \max_{1\leq i\leq d}|L(T,x,i)|$. It is known that Keller's conjecture is true in dimension seven for cube tilings $[0,1)^7+T$ for which $r^-(T)\leq 2$. In the present paper we show that it is also true for $d=7$ if $r^+(T)\geq 6$. Thus, if $[0,1)^d+T$ is a counterexample to Keller's conjecture in dimension seven, then $r^-(T),r^+(T)\in \{3,4,5\}$.