Discrete tomography: Determination of finite sets by X-rays

We study the determination of finite subsets of the integer lattice ${\Bbb Z}^n$, $n\ge 2$, by X-rays. In this context, an X-ray of a set in a direction $u$ gives the number of points in the set on each line parallel to $u$. For practical reasons, only X-rays in lattice directions, that is, directions parallel to a nonzero vector in the lattice, are permitted. By combining methods from algebraic number theory and convexity, we prove that there are four prescribed lattice directions such that convex subsets of ${\Bbb Z}^n$ (i.e., finite subsets $F$ with $F={\Bbb Z}^n\cap {\mathrm {conv}}\,F$) are determined, among all such sets, by their X-rays in these directions. We also show that three X-rays do not suffice for this purpose. This answers a question of Larry Shepp, and yields a stability result related to Hammer's X-ray problem. We further show that any set of seven prescribed mutually nonparallel lattice directions in ${\Bbb Z}^2$ have the property that convex subsets of ${\Bbb Z}^2$ are determined, among all such sets, by their X-rays in these directions. We also consider the use of orthogonal projections in the interactive technique of successive determination, in which the information from previous projections can be used in deciding the direction for the next projection. We obtain results for finite subsets of the integer lattice and also for arbitrary finite subsets of Euclidean space which are the best possible with respect to the numbers of projections used.