Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets

Abstract

We consider algebraic Delone sets $\Lambda$ in the Euclidean plane and address the problem of distinguishing convex subsets of $\Lambda$ by X-rays in prescribed $\Lambda$-directions, i.e., directions parallel to nonzero interpoint vectors of $\Lambda$. 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 $\Lambda$ there are four prescribed $\Lambda$-directions such that any two convex subsets of $\Lambda$ can be distinguished by the corresponding X-rays. We further prove the existence of a natural number $c_{\Lambda }$ such that any two convex subsets of $\Lambda$ can be distinguished by their X-rays in any set of $c_{\Lambda }$ prescribed $\Lambda$-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.

21 pages, 1 figure

References in corpus (4)

• Discrete Tomography of Planar Model Sets
• Discrete Tomography of Icosahedral Model Sets
• A Note on Affinely Regular Polygons
• On the Existence of $U$-Polygons of Class $c\ge 4$ in Planar Point Sets

Cited by in corpus (2)

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• Discrete tomography: Magic numbers for $N$-fold symmetry