论文标题
在某些方向和广义方向集的密度上
On denseness of certain direction and generalized direction sets
论文作者
论文摘要
Leonetti和Sanna最近引入的方向集是对正整数子集的比率集的概括。在本文中，我们概括了方向集的概念，并为正整数的子集定义了{\ it $ k $ generalized方向集}和{\ it独特的$ k $ generalized Directions集}。我们证明了$ \ Mathcal {s}^{k -1}子集的必要条件= 1 \} $被实现为独特的$ k $概括方向集的累积点集。我们为正整数的某些特定子集提供了足够的条件，以便相应的$ k $概括方向集在$ \ MATHCAL {s}^{k -1} $中密集。我们还考虑了某些方向集的密度属性，并对Leonetti和Sanna提出的问题给出了部分答案。最后，我们考虑了代数数字字段的框架中的一个类似问题。
Direction sets, recently introduced by Leonetti and Sanna, are generalization of ratio sets of subsets of positive integers. In this article, we generalize the notion of direction sets and define {\it $k$-generalized direction sets} and {\it distinct $k$-generalized direction sets} for subsets of positive integers. We prove a necessary condition for a subset of $\mathcal{S}^{k - 1} := \{\underline{x} \in [0,1]^{k} : ||\underline{x}|| = 1\}$ to be realized as the set of accumulation points of a distinct $k$-generalized direction set. We provide sufficient conditions for some particular subsets of positive integers so that the corresponding $k$-generalized direction sets are dense in $\mathcal{S}^{k - 1}$. We also consider the denseness properties of certain direction sets and give a partial answer to a question posed by Leonetti and Sanna. Finally we consider a similar question in the framework of an algebraic number field.
