Variants of the Erdos distinct sums problem and variance method

Let Sigma = {a(1), ..., a(n)} be a set of positive integers with a1 < ... < an such that all 2n subset sums are pairwise distinct. A famous conjecture of Erdos states that an > C2n for some constant C, while the best result known to date is of the form an > C 2n/root n. In this paper, we propose a generalization of the Erdos distinct sum problem that is in the same spirit as those of the Davenport and the Erd & odblac;s-Ginzburg-Ziv constants recently introduced in Caro et al. (2022) and in Caro and Schmitt (2022). More precisely, we require that the non-zero evaluations of the mth degree symmetric polynomial are all distinct over the subsequences of Sigma whose size is at most lambda n, for a given lambda is an element of (0, 1], considering Sigma as a sequence in Z(k) with each coordinate of each ai in [0, M]. If F-lambda,F-n denotes the family of subsets of [1, n] whose size is at most lambda n, our main result is that, for each k, m, and lambda, there exists an explicit constant C(k,m,lambda )such that M >= C-k,C-m,C-lambda (1 +o(1))|F lambda,n| 1/mk / n(1- 1/ 2m) (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.