Translational tilings by a polytope, with multiplicity

Gravin, Nick; Robins, Sinai; Shiryaev, Dmitry

doi:10.48550/arxiv.1103.3163

Cited by 3 publications

(7 citation statements)

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“…Proof. We know from [6] (see Lemma 3.1 and Lemma 3.2 in [6]) that if P tiles with Λ and µ is a leg measure on P then µ also tiles with Λ, at level 0. In other words µ * δ Λ = 0.…”

Section: Preliminariesmentioning

confidence: 99%

“…It was already shown in [6] that a multiple tiler in R 3 must be a zonotope, i. e. a Minkowski sum of line segments. Here we will show that given the non-discreteness of supp δ Λ , we can deduce that a zonotope is a Minkowski sum of two 2-dimensional symmetric polygons.…”

Section: The Intersection Property Implies Quasi-periodicity Of λmentioning

confidence: 99%

“…Theorem (Gravin, Robins, Shiryaev 2011 [6]). If a convex polytope k-tiles R d by translations, then it is centrally symmetric and its facets are centrally symmetric.…”

Section: Introductionmentioning

confidence: 99%

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Structure Results for Multiple Tilings in 3D

Gravin

Kolountzakis

Robins

et al. 2013

Discrete Comput Geom

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“…Proof. We know from [6] (see Lemma 3.1 and Lemma 3.2 in [6]) that if P tiles with Λ and µ is a leg measure on P then µ also tiles with Λ, at level 0. In other words µ * δ Λ = 0.…”

Section: Preliminariesmentioning

confidence: 99%

Section: The Intersection Property Implies Quasi-periodicity Of λmentioning

confidence: 99%

See 1 more Smart Citation

Structure Results for Multiple Tilings in 3D

Gravin

Kolountzakis

Robins

et al. 2013

Discrete Comput Geom

Self Cite

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“…In this case full-dimensional case, we have to take care of those ξ ∈ R d which are orthogonal to all of P , but this implies that there is only the 0-dimensional subspace here, namely ξ = 0. Also, noting that in this first full-dimensional step we have simply Proj P (ξ) = ξ, (27) becomes…”

Section: Proof Of the Combinatorial Stokes Formula For Pmentioning

confidence: 99%

“…In the field of tiling and multi-tiling, solid angles play a role [27] in giving an equivalent condition for a rational polytope P to be able to multi-tile R d by translations with a lattice (and more general sets). This condition is essentially equivalent to saying that the sum of the solid angles of P , taken at all integer points, equals the volume of P , and this hold for all translations of P .…”

Section: Further Remarksmentioning

confidence: 99%

Fourier transforms of polytopes, solid angle sums, and discrete volume

Diaz¹,

Le²,

Robins³

2016

Preprint

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Given a real closed polytope P , we first describe the Fourier transform of its indicator function by using iterations of Stokes' theorem. We then use the ensuing Fourier transform formulations, together with the Poisson summation formula, to give a new algorithm to count fractionally-weighted lattice points inside the one-parameter family of all real dilates of P . The combinatorics of the face poset of P plays a central role in the description of the Fourier transform of P . We also obtain a closed form for the codimension-1 coefficient that appears in an expansion of this sum in powers of the real dilation parameter t. This closed form generalizes some known results about the Macdonald solid-angle polynomial, which is the analogous expression traditionally obtained by requiring that t assumes only integer values. Although most of the present methodology applies to all real polytopes, a particularly nice application is to the study of all real dilates of integer (and rational) polytopes.

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Lattice zonotopes of degree 2

Beck¹,

Ellinor²,

Jochemko³

2022

Preprint

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The Ehrhart polynomial ehr P (n) of a lattice polytope P gives the number of integer lattice points in the n-th dilate of P for all integers n ≥ 0. The degree of P is defined as the degree of its h * -polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory.A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree 2 thereby complementing results of Scott (1976), Treutlein (2010), and Henk-Tagami (2009. Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all 3-dimensional zonotopes of degree 2.

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Translational tilings by a polytope, with multiplicity

Cited by 3 publications

References 0 publications

Structure Results for Multiple Tilings in 3D

Structure Results for Multiple Tilings in 3D

Fourier transforms of polytopes, solid angle sums, and discrete volume

Lattice zonotopes of degree 2

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