Higher degree Erdős distinct evaluations problem

Abstract

Let $\Sigma=\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 c\cdot 2^n$ 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 sub-sequences of $\Sigma$. Even though these evaluations can not be seen as the values assumed by the sum of independent random variables, surprisingly, the variance method works to provide a nontrivial lower bound on $a_n$. Indeed, the main result here presented is to show that $$a_n>c_m\cdot 2^{\frac{n}{m}}/n^{1-\frac{1}{2m}}.$$