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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2207.00053 |
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| _version_ | 1866914233397018624 |
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| author | Xu, Zixiang Yip, Chi Hoi |
| author_facet | Xu, Zixiang Yip, Chi Hoi |
| contents | Given a finite abelian group $G$ and a subset $J\subset G$ with $0\in J$, let $D_{G}(J,N)$ be the maximum size of $A\subset G^{N}$ such that the difference set $A-A$ and $J^{N}$ have no non-trivial intersection. Recently, this extremal problem has been widely studied for different groups $G$ and subsets $J$. In this paper, we generalize and improve the relevant results by Alon and by Hegedűs by building a bridge between this problem and cyclotomic polynomials with the help of algebraic graph theory. In particular, we construct infinitely many non-trivial families of $G$ and $J$ for which the current known upper bounds on $D_{G}(J, N)$ can be improved exponentially. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_00053 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Intersective sets over abelian groups Xu, Zixiang Yip, Chi Hoi Combinatorics Number Theory 05D05, 11B30, 11C08 Given a finite abelian group $G$ and a subset $J\subset G$ with $0\in J$, let $D_{G}(J,N)$ be the maximum size of $A\subset G^{N}$ such that the difference set $A-A$ and $J^{N}$ have no non-trivial intersection. Recently, this extremal problem has been widely studied for different groups $G$ and subsets $J$. In this paper, we generalize and improve the relevant results by Alon and by Hegedűs by building a bridge between this problem and cyclotomic polynomials with the help of algebraic graph theory. In particular, we construct infinitely many non-trivial families of $G$ and $J$ for which the current known upper bounds on $D_{G}(J, N)$ can be improved exponentially. |
| title | Intersective sets over abelian groups |
| topic | Combinatorics Number Theory 05D05, 11B30, 11C08 |
| url | https://arxiv.org/abs/2207.00053 |