Tilings of convex polyhedral cones and topological properties of self-affine tiles

Abstract

Let ${\text{a}}_1,\dots ,{\text{a}}_r$ be vectors in a half-space of ${\mathbb{R}}^n$. We call $$C={\text{a}}_1{\mathbb{R}}^{+}+\cdots +{\text{a}}_r{\mathbb{R}}^{+}$$ a convex polyhedral cone, and call $\{{\text{a}}_1,\dots ,{\text{a}}_r\}$ a generator set of $C$. A generator set with the minimal cardinality is called a frame. We investigate the translation tilings of convex polyhedral cones. Let $T\subset {\mathbb{R}}^n$ be a compact set such that $T$ is the closure of its interior, and $\mathcal{J}\subset {\mathbb{R}}^n$ be a discrete set. We say $(T,\mathcal{J})$ is a translation tiling of $C$ if $T+\mathcal{J}=C$ and any two translations of $T$ in $T+\mathcal{J}$ are disjoint in Lebesgue measure. We show that if the cardinality of a frame of $C$ is larger than $\dim C$, the dimension of $C$, then $C$ does not admit any translation tiling; if the cardinality of a frame of $C$ equals $\dim C$, then the translation tilings of $C$ can be reduced to the translation tilings of $({\mathbb{Z}}^{+}{)}^n$. As an application, we characterize all the self-affine tiles possessing polyhedral corners, which generalizes a result of Odlyzko [A. M. Odlyzko, \textit{Non-negative digit sets in positional number systems}, Proc. London Math. Soc., \textbf{37}(1978), 213-229.].