Tilings of Convex Polyhedral Cones and Topological Properties of Self-Affine Tiles
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Tilings of Convex Polyhedral Cones and Topological Properties of Self-Affine Tiles Ya-min Yang1 · Yuan Zhang2 Received: 2 August 2019 / Revised: 21 June 2020 / Accepted: 10 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let a1 , . . . , ar be vectors in a half-space of Rn . We call C = a1 R+ + · · · + ar R+ a convex polyhedral cone and {a1 , . . . , ar } 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 ⊂ Rn be a compact set such that T is the closure of its interior, and J ⊂ Rn 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)). Keywords Convex polyhedral cone · Translation tiling · Self-affine tile Mathematics Subject Classification 52C22 · 51M20
Editor in Charge: Kenneth Clarkson This work is supported by NSFC Nos. 11431007, 11601172, 11971195, 12071167, and Fundamental Research Funds for Central Universities No. 2662015PY217, and Self-Determined Research Funds of CCNU from the Colleges’ Basic Research and Operation of MOE under Grant CCNU17XJ034. Ya-min Yang [email protected] Yuan Zhang [email protected] 1
Institute of Applied Mathematics, College of Science, Huazhong Agricultural University, Wuhan 430070, China
2
Department of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
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Discrete & Computational Geometry
1 Introduction Let a1 , a2 , . . . , am be m non-zero vectors in a half-space of Rn , that is, there is a non-zero vector β ∈ Rn such that the inner product a j , β > 0 for all j = 1, . . . , m. Denote R+ = {x ∈ R : x ≥ 0}. We call the set of all non-negative combinations of these vectors, C = a1 R+ + · · · + am R+ = {λ1 a1 + · · · + λm am : all λi ≥ 0}, a convex polyhedral cone. In this case, we also say C is spanned by {a1 , . . . , am }. The convex polyhedral cone is an important object in convex analysis, see, for instance, Rockafellar [19]. The main purpose of the present paper is to characterize the translation tilings of convex polyhedral cones. Definition 1.1 Let X ⊂ Rn , T ⊂ Rn be a compact set, and J ⊂ Rn be a (finite or infinite) discrete set. We say that (T , J ) is a packing of X if T + J ⊂ X , and T + t1 and T + t2 are disjoint in Lebesgue measure for any t1 = t2 ∈ J . (T , J ) is calle
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