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 u 1 , … , u k 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 ) ⩽ 2 k − 1 . We establish a mildly exponential lower bound F ( k ) > 2 ( k / 2 ) 1 / 3 , and we improve the upper bound to F ( k ) ⩽ O ( 1.81712 k ) (or O ( 1.79964 k ) if we allow A to be a multiset).