PUB - Publikationen an der Universität Bielefeld

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 lines through two different points 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.