Variations on the Erdős distinct-sums problem.

Let { a 1 ,... , a n } be a set of positive integers with a 1 < ⋯ < a n such that all 2 n subset sums are distinct. A famous conjecture by Erdős states that a n > c ⋅ 2 n for some constant c , while the best result known to date is of the form a n > c ⋅ 2 n / n . In this paper, inspired by an information-theoretic interpretation, we extend the study to vector-valued elements a i ∈ Z k and we weaken the condition by requiring that only sums corresponding to subsets of size smaller than or equal to λ n be distinct. For this case, we derive lower and upper bounds on the smallest possible value of a n. [ABSTRACT FROM AUTHOR]