Structure theorem for multiple addition and the Frobenius problem

Let A subset of or equal to [0; l] be a set of n integers, and let h greater than or equal to 2. By how much does \hA\ exceed \(h-1)A\? How can one estimate \hA\ in terms of n, l? We give sharp lower bounds extending and generalizing the well-known theorem of Freiman for \2A\. A number of applications are provided as well. In particular, we give a solution for the old extremal problem of Frobenius-Erdos-Graham concerning estimating of the largest integer, non-representable by a linear form. In a sense, our solution can not be improved. (C) 1996 Academic Press, Inc.