论文标题
整数和整体晶格中的无限共同最小对
Infinite co-minimal pairs in the integers and integral lattices
论文作者
论文摘要
给定了两个非空的子集$ a，b $的一组$ g $，据说如果$ a \ cdot b = g $，以及$ a'\ cdot b \ cdot b \ subsetneq g $对于任何$ \ emberyset \ neq a'\ neq a'\ subsetneq a $ and $ a \ cdot b'\ cdot b'\ subsete for任何$ \ subsetneq b $。在本文中，我们在整数和整体晶格中展示了几个新结果。我们证明，对于任何$ d \ geq 1 $，组$ \ mathbb {z}^{2d} $承认，对于每种自动形态$σ$，都存在一个子集$ a $ a $ a $ a $ \ mathbb {z}^{2d^co-min $ a $ a $ a $ a $ a（a）在整数中的共同最小对的存在和构建与无限的无限基数的子集$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a和$ b $（$ a \ neq b $）的无限基数尚不清楚。我们表明，这种对存在并明确构造这些对满足许多代数特性的对。
Given two nonempty subsets $A, B$ of a group $G$, they are said to form a co-minimal pair if $A \cdot B = G$, and $A' \cdot B \subsetneq G$ for any $\emptyset \neq A' \subsetneq A$ and $A\cdot B' \subsetneq G$ for any $\emptyset \neq B' \subsetneq B$. In this article, we show several new results on co-minimal pairs in the integers and the integral lattices. We prove that for any $d\geq 1$, the group $\mathbb{Z}^{2d}$ admits infinitely many automorphisms such that for each such automorphism $σ$, there exists a subset $A$ of $\mathbb{Z}^{2d}$ such that $A$ and $σ(A)$ form a co-minimal pair. The existence and construction of co-minimal pairs in the integers with both the subsets $A$ and $B$ ($A\neq B$) of infinite cardinality was unknown. We show that such pairs exist and explicitly construct these pairs satisfying a number of algebraic properties.
