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

作者：Yang, Ya-min^*; Zhang, Yuan

来源：Discrete and Computational Geometry, 2021, 66(3): 876-901.

DOI：10.1007/s00454-020-00249-1

• Convex polyhedral cone
• Translation tiling
• Self-affine tile

Let a(1),..., a(r) be vectors in a half-space of R-n. We call C = a(1)R(+) + ... + a(r)R(+) a convex polyhedral cone and {a(1), ..., 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 of R-n be a compact set such that T is the closure of its interior, and J subset of R-n be a discrete set. We say (T, J) is a translation tiling of C if T + J = C and any two translations of T in T + 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 (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)).

• 单位
  华中农业大学