Some Observations on Subset Sum Representations

Moulton and Develin have investigated the notion of representing various sets S of positive integers, of size m say, as subset sums of smaller sets. The rank of a given set S is the smallest positive integer rk(S) ≤ m such that S is represented by an integer set of size rk(S). In this note, we primarily consider sets of the form {1, 2, 3, . . .} for positive integer m ≥ 2. Given a positive integer k, we ask for the smallest M such that {1, 2, 3, . . . , k} is independent for all m ≥ M , and provide some answers. We then use a result of Sprague to show that any nondecreasing positive integer sequence a = {a1, a2, . . .} that grows polynomially, and in particular the set {1, 2, 3, . . .} for fixed exponent m, has limiting rank zero.