On subset-sum-distinct sequences of positive integers

An SSD-sequence of integers is one in which each subset is uniquely determined by its sum. Such sequences are "sparse". Ryavec used a generating function technique to show that the sum of the reciprocals of the terms of such a sequence is at most two, and that the greedy algorithm generates the unique extremal sequence. Here his result is obtained by elementary "Karamata-type" inequalities that are shown to have a wide range of applicability to many related problems. Included is an elementary proof of the theorem of Steele, Hanson, and Stenger. In addition to many variations on the original result of Ryavec, a general compactness result for problems of this sort is established. The most intricate results of this paper concern SSD-sequences with congruence conditions on the subset sums. Here a detailed analysis shows that the greedy algorithm is optimal infinitely often, but also fails to be optimal infinitely often. The famous open question of the optimality of the Conway-Guy sequence is not resolved, but an elementary method of L. Moser bearing on this is shown to be related to Laplace's method for the asymptotic estimation of certain integrals.U of I OnlyETDs are only available to UIUC Users without author permissio