Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets
-
Christian Huck
and
Michael Spieß
Abstract
We consider algebraic Delone sets Λ in the Euclidean plane and address the problem of distinguishing convex subsets of Λ by X-rays in prescribed Λ-directions, i.e. directions parallel to lines through two different points of Λ. Here, an X-ray in direction u of a finite set gives the number of points in the set on each line parallel to u. It is shown that for any algebraic Delone set Λ there are four prescribed Λ-directions such that any two convex subsets of Λ can be distinguished by the corresponding X-rays. We further prove the existence of a natural number cΛ such that any two convex subsets of Λ can be distinguished by their X-rays in any set of cΛ prescribed Λ-directions. In particular, this extends a well-known result of Gardner and Gritzmann on the corresponding problem for planar lattices to nonperiodic cases that are relevant in quasicrystallography.
©[2013] by Walter de Gruyter Berlin Boston
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- K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space, II: A structure theorem for r(M) > 10
- Groups of positive weighted deficiency and their applications
- The Witt group of non-degenerate braided fusion categories
- Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties
- Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets
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Articles in the same Issue
- Essential dimension of algebraic tori
- K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space, II: A structure theorem for r(M) > 10
- Groups of positive weighted deficiency and their applications
- The Witt group of non-degenerate braided fusion categories
- Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties
- Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets
- Brauer group of moduli of principal bundles over a curve
