Variants of the Erd\H{o}s distinct sums problem and variance method

Let $\Sigma=\{a_1, \ldots , a_n\}$ be a set of positive integers with $a_1 < \ldots < a_n$ such that all $2^n$ subset sums are pairwise distinct. A famous conjecture of Erd\H{o}s 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 Erd\H{o}s distinct sum problem that is in the same spirit as those of the Davenport and the Erd\H{o}s-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 $$M\geq C_{k,m,\lambda} \frac{(1+o(1)) |\mathcal{F}_{\lambda,n}|^{\frac{1}{mk}}}{n^{1 - \frac{1}{2m}}}.$$