Variants of the Erdős distinct sums problem and variance method

Abstract

Let $\Sigma =\{a_1,\dots ,a_n\}$ be a set of positive integers with $a_1<\dots <a_n$ such that all $2^n$ subset sums are pairwise distinct. A famous conjecture of Erdős states that $a_n>C\cdot 2^n$ for some constant $C$, while the best result known to date is of the form $a_n>C\cdot 2^n/\sqrt{n}$. In this paper, we propose a generalization of the Erdős distinct sum problem that is in the same spirit as those of the Davenport and the Erdős–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 $m$th degree symmetric polynomial are all distinct over the subsequences of $\Sigma$ whose size is at most $\lambda n$, for a given $\lambda \in (0,1]$, considering $\Sigma$ as a sequence in $Z^k$ with each coordinate of each $a_i$ in $[0,M]$. If $F_{\lambda ,n}$ 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 $M\ge C_{k,m,\lambda }\frac{(1+o\left(1\right)){\left|F_{\lambda ,n}\right|}^{\frac{1}{mk}}}{n^{1-\frac{1}{2m}}}.$