The smallest sets of points not determined by their X-rays

zakiya;

Let F be an n-point set in Kd with K ∈ {R, Z} and d >= 2. A (discrete) X-ray of F in direction s gives the number of points of F on each line parallel to s. We define ψKd (m) as the minimum number n for which there exist m directions s1, . . . , sm (pairwise linearly independent and
spanning Rd) such that two n-point sets in Kd exist that have the same X-rays in these directions. The bound ψZd (m) <= 2m−1 has been observed many times in the literature. In this note, we show ψKd (m) = O(md+1+ε) for ε > 0. For the cases Kd = Zd and Kd = Rd, d > 2, this represents the first upper bound on ψKd (m) that is polynomial in m. As a corollary, we derive bounds on the sizes of solutions to both the classical and two-dimensional Prouhet–Tarry–Escott problem. Additionally, we establish lower bounds on ψKd that enable us to prove a strengthened version of R´enyi’s theorem for points in Z2.