Concentrating Subset Sums at k Points

We consider the problem of maximimizing, over all choices of n nonzero elements a 1 ; : : : ; a n 2 R m , the number of the 2 n subset sums P i2I a i , over all index sets I, belonging to some specified target set T . M. Miller, Roberts, and Simpson investigated the case m = 1 and T = f0; 1g of this problem, and showed that the maximum in their case is \Gamma n+1 b n+1 2 c \Delta , but it remained open until now to prove the essential uniqueness of the extremal solutions a i that achieve this maximum. More generally, we determine the maximum, as well as solutions achieving it, over n arbitrary elements a i and target sets T of k arbitrary points in R m . We also obtain the same maximum number of sums when T is a union of k open balls of diameter min i ja i j