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

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