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

Let F be an n-point set in K-d with K is an element of {R, Z} and d >= 2. A (discrete) X-ray of F in direction s gives the munber of points of P on each line parallel to 8. We define psi K-d (m) as the minimum number n for which there exist m directions s(1), ..., s(m) (pairwise linearly independent and spanning R-d) such that two n-point sets in K-d exist that have the same X-rays in these directions. The bound psi(Z)d/(m) <= 2(m-1) has been observed many times in the literature. In this note, we show psi K-d(m) = O(m(d+1+epsilon)) for epsilon > 0. For the cases K-d = Z(d) and K-d = R-d d > 2, this represei its the first upper hound on psi K-d(m) that is polynomial in m. As a corollary, We derive hounds on the sizes of solutions to both the classical and two-dimensional Prouhet Tarry Escott problem. Additionally, we establish lower bounds on psi K-d that enable us to prove a strengthened version of Renyi's theorem for points in Z(2).