How many points can be reconstructed from k projections?

Let A be an n-point set in the plane. A discrete X-ray of A in direction u gives the number of points of A on each line parallel to u. We define F(k) as the maximum number n such that there exist k directions u1,…,uk such that every set of at most n points in the plane can be uniquely reconstructed from its discrete X-rays in these directions. A simple “cube” construction shows F(k)⩽2k−1. We establish a mildly exponential lower bound F(k)>(k/2)1/32, and we improve the upper bound to F(k)⩽O(k1.81712) (or O(k1.79964) if we allow A to be a multiset).