Variants of the Erdős distinct sums problem and variance method

Abstract

Let $Σ=\{a_1, \ldots , a_n\}$ 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 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 $Σ$ whose size is at most $λn$, for a given $λ\in (0,1]$, considering $Σ$ as a sequence in $\mathbb{Z}^k$ with each coordinate of each $a_i$ in $[0,M]$. If $\mathcal{F}_{λ,n}$ denotes the family of subsets of $[1,n]$ whose size is at most $λn$, our main result is that, for each $k,m,$ and $λ$, there exists an explicit constant $C_{k,m,λ}$ such that $$ M\geq C_{k,m,λ} \frac{(1+o(1)) |\mathcal{F}_{λ,n}|^{\frac{1}{mk}}}{n^{1 - \frac{1}{2m}}}.$$

Theorems & Definitions (20)

• Theorem 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Corollary 3.4
• Lemma 3.5
• Theorem 3.6
• Remark 3.7
• Remark 3.8
• Lemma 4.1: Schwartz-Zippel Lemma