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 \cite{CGS} and in \cite{CS}. 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 ${\mathbb{Z}}^k$ with each coordinate of each $a_i$ in $[0,M]$. If ${\mathcal{F}}_{\lambda ,n}$ 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 $$M\ge C_{k,m,\lambda }\frac{(1+o(1))|{\mathcal{F}}_{\lambda ,n}{|}^{\frac{1}{mk}}}{n^{1-\frac{1}{2m}}}.$$