$\{\pm 1\}$-weighted zero-sum constants
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915842542796800 |
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| author | Paul, Krishnendu Paul, Shameek |
| author_facet | Paul, Krishnendu Paul, Shameek |
| contents | Let $A,B\subseteq \mathbb Z_n\setminus\{0\}$. A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_n$ is called an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that $a_1x_1+\cdots+a_kx_k=0$ and $b_1a_1+\cdots+b_ka_k=0$. The constant $E_{A,B}(n)$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in $\mathbb Z_n$ has an $(A,B)$-weighted zero-sum subsequence of length $n$. We determine the constant $E_{A,B}(n)$ and the related constants $C_{A,B}(n)$ and $D_{A,B}(n)$ when $A=\{\pm 1\}$ and $B=\{1\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07251 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $\{\pm 1\}$-weighted zero-sum constants Paul, Krishnendu Paul, Shameek Number Theory 11B50, 11B75 Let $A,B\subseteq \mathbb Z_n\setminus\{0\}$. A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_n$ is called an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that $a_1x_1+\cdots+a_kx_k=0$ and $b_1a_1+\cdots+b_ka_k=0$. The constant $E_{A,B}(n)$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in $\mathbb Z_n$ has an $(A,B)$-weighted zero-sum subsequence of length $n$. We determine the constant $E_{A,B}(n)$ and the related constants $C_{A,B}(n)$ and $D_{A,B}(n)$ when $A=\{\pm 1\}$ and $B=\{1\}$. |
| title | $\{\pm 1\}$-weighted zero-sum constants |
| topic | Number Theory 11B50, 11B75 |
| url | https://arxiv.org/abs/2603.07251 |