论文标题
相互作用的亚伯集团
Intersective sets over abelian groups
论文作者
论文摘要
给定有限的Abelian Group $ G $和一个子集$ J \ subset G $，$ 0 \ in J $，让$ d_ {g}（j，j，n）$是$ a \ subset g^{n} $的最大大小，以使差异集$ a-a-a $ and $ a-a $和$ j^{n} $没有非trivial intersection。最近，对不同组的$ g $和subset $ j $进行了广泛研究这个极端问题。在本文中，我们通过代数图理论的帮助，通过在该问题和环形多项式之间建立桥梁来概括和改善Heads的相关结果。特别是，我们构建了$ g $和$ j $的许多非平凡的家庭，为$ d_ {g}（j，n）$上的当前已知上限可以呈指数级改进。我们还获得了一个新的上限$ d _ {\ mathbb {f} _ {p}}（\ {0,1 \}，n），n）\ le（\ frac {1} {2} {2}+o（1））（p-1））（p-1））（p-1）^{n} $，这改善了先前最佳的结果byhuang，klang，kluran，＆pohain。
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. We also obtain a new upper bound $D_{\mathbb{F}_{p}}(\{0,1\},N)\le (\frac{1}{2}+o(1))(p-1)^{N}$, which improves the previously best-known result by Huang, Klurman, and Pohoata.
