$Q_p$-weighted zero-sum constants
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908776534114304 |
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| author | Paul, Krishnendu Paul, Shameek |
| author_facet | Paul, Krishnendu Paul, Shameek |
| contents | A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_p$ is called a $(Q_p,\mathbf 1)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in Q_p$ such that $a_1x_1+\cdots+a_kx_k=0$ and $a_1+\cdots+a_k=0$. The constant $E_{Q_p,\mathbf 1}$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in $\mathbb Z_p$ has a $(Q_p,\mathbf 1)$-weighted zero-sum subsequence of length $p$. We determine the constant $E_{Q_p,\mathbf 1}$ and the related constants $C_{Q_p,\mathbf 1}$ and $D_{Q_p,\mathbf 1}$. We also study some $(Q_p,B)$-weighted zero-sum constants where $B$ is a subset of $Q_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14122 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $Q_p$-weighted zero-sum constants Paul, Krishnendu Paul, Shameek Number Theory Combinatorics 11B50, 11B75 A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_p$ is called a $(Q_p,\mathbf 1)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in Q_p$ such that $a_1x_1+\cdots+a_kx_k=0$ and $a_1+\cdots+a_k=0$. The constant $E_{Q_p,\mathbf 1}$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in $\mathbb Z_p$ has a $(Q_p,\mathbf 1)$-weighted zero-sum subsequence of length $p$. We determine the constant $E_{Q_p,\mathbf 1}$ and the related constants $C_{Q_p,\mathbf 1}$ and $D_{Q_p,\mathbf 1}$. We also study some $(Q_p,B)$-weighted zero-sum constants where $B$ is a subset of $Q_p$. |
| title | $Q_p$-weighted zero-sum constants |
| topic | Number Theory Combinatorics 11B50, 11B75 |
| url | https://arxiv.org/abs/2601.14122 |