Variations on the Erd?s distinct-sums problem

Let {a1, ..., an} be a set of positive integers with a1 < middot middot middot < an such that all 2n subset sums are distinct. A famous conjecture by Erdos states that an > c middot2n for some constant c, while the best result known to date is of the form an > c middot 2n/,/n. In this paper, inspired by an information-theoretic interpretation, we extend the study to vectorvalued elements ai E Zk and we weaken the condition by requiring that only sums corresponding to subsets of size smaller than or equal to lambda n be distinct. For this case, we derive lower and upper bounds on the smallest possible value of an.(c) 2022 Elsevier B.V. All rights reserved.