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

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