Quasi-periodic Tiling with Multiplicity: A Lattice Enumeration Approach
Quasi-periodic Tiling with Multiplicity: A Lattice Enumeration Approach
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具有多重性的准周期平铺：一种格子枚举方法
DOI：
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发表时间：
2014
影响因子：
0.8
通讯作者：
Swee Hong Chan
中科院分区：
文献类型：
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作者：
Swee Hong Chan
The k-tiling problem for a convex polytope P is the problem of covering $$ mathbb {R}^d$$Rd with translates of P using a discrete multiset $$varLambda $$Λ of translation vectors such that every point in $$ mathbb {R}^d$$Rd is covered exactly k times, except possibly for the boundary of P and its translates. A classical result in the study of tiling problems is a theorem of McMullen [Mathematika 27(1):113–121, 1980] that a convex polytope P that 1-tiles $$ mathbb {R}^d$$Rd with a discrete multiset $$varLambda $$Λ can, in fact, 1-tile $$ mathbb {R}^d$$Rd with a lattice $$mathcal {L}$$L. A generalization of McMullen’s theorem for k-tiling was conjectured by Gravin et al. [Combinatorica 32(6):629–649, 2012], which states that if Pk-tiles $$ mathbb {R}^d$$Rd with a discrete multiset $$varLambda $$Λ, then Pm-tiles $$ mathbb {R}^d$$Rd with a lattice $$mathcal {L}$$L for some m. In this paper, we consider the case when Pk-tiles $$ mathbb {R}^d$$Rd with a discrete multiset $$varLambda $$Λ such that every element of $$varLambda $$Λ is contained in a quasi-periodic set $$mathcal {Q}$$Q (i.e., a finite union of translated lattices). This is motivated by the result of Gravin et al. [Discrete Comput Geom 50(4):1033–1050, 2013] and Kolountzakis [Discrete Comput Geom 23(4):537–553, 2000], showing that for $$d in {2,3}$$d∈{2,3}, if a polytope Pk-tiles $$ mathbb {R}^d$$Rd with a discrete multiset $$varLambda $$Λ, then Pm-tiles $$ mathbb {R}^d$$Rd with a quasi-periodic set $$mathcal {Q}$$Q for some m. Here we show for all values of d that if a polytope Pk-tiles $$ mathbb {R}^d$$Rd with a discrete multiset $$varLambda $$Λ that is contained in a quasi-periodic set $$mathcal {Q}$$Q that satisfies a mild hypothesis, then Pm-tiles $$ mathbb {R}^d$$Rd with a lattice $$mathcal {L}$$L for some m. This strengthens the results of Gravin, Kolountzakis, Robins, and Shiryaev, and is a step in the direction of proving the conjecture of Gravin et al. [Combinatorica 32(6):629–649, 2012].
The k-tiling problem for a convex polytope P is the problem of covering $$ mathbb {R}^d$$Rd with translates of P using a discrete multiset $$varLambda $$Λ of translation vectors such that every point in $$ mathbb {R}^d$$Rd is covered exactly k times, except possibly for the boundary of P and its translates. A classical result in the study of tiling problems is a theorem of McMullen [Mathematika 27(1):113–121, 1980] that a convex polytope P that 1-tiles $$ mathbb {R}^d$$Rd with a discrete multiset $$varLambda $$Λ can, in fact, 1-tile $$ mathbb {R}^d$$Rd with a lattice $$mathcal {L}$$L. A generalization of McMullen’s theorem for k-tiling was conjectured by Gravin et al. [Combinatorica 32(6):629–649, 2012], which states that if Pk-tiles $$ mathbb {R}^d$$Rd with a discrete multiset $$varLambda $$Λ, then Pm-tiles $$ mathbb {R}^d$$Rd with a lattice $$mathcal {L}$$L for some m. In this paper, we consider the case when Pk-tiles $$ mathbb {R}^d$$Rd with a discrete multiset $$varLambda $$Λ such that every element of $$varLambda $$Λ is contained in a quasi-periodic set $$mathcal {Q}$$Q (i.e., a finite union of translated lattices). This is motivated by the result of Gravin et al. [Discrete Comput Geom 50(4):1033–1050, 2013] and Kolountzakis [Discrete Comput Geom 23(4):537–553, 2000], showing that for $$d in {2,3}$$d∈{2,3}, if a polytope Pk-tiles $$ mathbb {R}^d$$Rd with a discrete multiset $$varLambda $$Λ, then Pm-tiles $$ mathbb {R}^d$$Rd with a quasi-periodic set $$mathcal {Q}$$Q for some m. Here we show for all values of d that if a polytope Pk-tiles $$ mathbb {R}^d$$Rd with a discrete multiset $$varLambda $$Λ that is contained in a quasi-periodic set $$mathcal {Q}$$Q that satisfies a mild hypothesis, then Pm-tiles $$ mathbb {R}^d$$Rd with a lattice $$mathcal {L}$$L for some m. This strengthens the results of Gravin, Kolountzakis, Robins, and Shiryaev, and is a step in the direction of proving the conjecture of Gravin et al. [Combinatorica 32(6):629–649, 2012].