Translational tilings by a polytope, with multiplicity


We study the problem of covering ℤd by overlapping translates of a convex polytope, such that almost every point of ℤd is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov [5] and Minkowski [15], and was later extended by Venkov and McMullen to give a complete characterization of all convex objects that 1-tile ℤd.

By contrast, for k ≥2, the collection of polytopes that k-tile is much wider than the collection of polytopes that 1-tile, and there is currently no known analogous characterization for the polytopes that k-tile. Here we first give the necessary conditions for polytopes P that k-tile, by proving that if P k-tiles ℤd by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski’s conditions for 1-tiling polytopes, but it turns out that very new methods are necessary for the development of the theory. In the case that P has rational vertices, we also prove that the converse is true; that is, if P is a rational polytope, is centrally symmetric, and has centrally symmetric facets, then P must k-tile ℤd for some positive integer k.

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Correspondence to Sinai Robins.

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The second author is grateful for the support of the Ministry of Education, Singapore, research grant MOE2011-T2-1-090

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Gravin, N., Robins, S. & Shiryaev, D. Translational tilings by a polytope, with multiplicity. Combinatorica 32, 629–649 (2012).

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Mathematics Subject Classification (2000)

  • 52B11
  • 52C07
  • 52C17
  • 52C22