Variants of the Erdős distinct sums problem and variance method
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911769198329856 |
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| author | Costa, Simone Della Fiore, Stefano Ferraguti, Andrea |
| author_facet | Costa, Simone Della Fiore, Stefano Ferraguti, Andrea |
| contents | 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 $$ M\geq C_{k,m,λ} \frac{(1+o(1)) |\mathcal{F}_{λ,n}|^{\frac{1}{mk}}}{n^{1 - \frac{1}{2m}}}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00642 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Variants of the Erdős distinct sums problem and variance method Costa, Simone Della Fiore, Stefano Ferraguti, Andrea Combinatorics Number Theory 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 $$ M\geq C_{k,m,λ} \frac{(1+o(1)) |\mathcal{F}_{λ,n}|^{\frac{1}{mk}}}{n^{1 - \frac{1}{2m}}}.$$ |
| title | Variants of the Erdős distinct sums problem and variance method |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2402.00642 |