Tilings of Convex Polyhedral Cones and Topological Properties of Self-Affine Tiles

Abstract

Let $${\varvec{a}}_1,\ldots , {\varvec{a}}_r$$ be vectors in a half-space of $${\mathbb {R}}^n$$ . We call $$C={\varvec{a}}_1{\mathbb {R}}^{+}+\cdots +{\varvec{a}}_r {\mathbb {R}}^{+}$$ a convex polyhedral cone and $$\{{\varvec{a}}_1,\ldots , {\varvec{a}}_r\}$$ 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 the dimension of C, then C does not admit any translation tiling; if the cardinality of a frame of C equals the dimension of 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 (that is, there exists a point of the tile such that a neighborhood of the point is congruent to a neighborhood of the vertex of a convex polyhedral cone), which generalizes a result of Odlyzko (Proc. Lond. Math. Soc. 37, 213–229 (1978)).