The smallest sets of points not determined by their X‐rays. (29th December 2014) - Alpers, Andreas; Larman, David G.
Abstract : LetF be ann ‐point set inK d withK ∈ R, Z andd ⩾ 2 . A (discrete) X‐ray ofF in directions gives the number of points ofF on each line parallel tos . We defineψ K d ( m ) as the minimum numbern for which there existm directionss 1, …, s m (pairwise linearly independent and spanningR d ) such that twon ‐point sets inK d exist that have the same X‐rays in these directions. The boundψ Z d ( m ) ⩽ 2 m − 1 has been observed many times in the literature. In this note, we showψ K d ( m ) = O ( m d + 1 + ε ) forε > 0 . For the casesK d = Z d andK d = R d, d > 2, this represents the first upper bound onψ K d ( m ) that is polynomial inm . 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 onψ K d that enable us to prove a strengthened version of Rényi's theorem for points inZ 2 .