On consecutive subset sums

We consider the subset sums analog of the linear Diophantine problem of Frobenius. It is shown that if A subset of or equal to [1; l] is a sufficiently dense set of n positive integers, then [2l-2n + 1; sigma - (2l - 2n + 1)] subset of or equal to A*, where sigma is the sum of all elements of A, and A* is the set of all subset sums of A. The interval above is best possible and cannot be extended. (C) 1998 Elsevier Science B.V. All rights reserved.