格点的多重可见性
Multiple Visibility of Lattice Points
DOI: 10.12677/PM.2021.112036, PDF,    国家自然科学基金支持
作者: 黄 旸：青岛大学数学与统计学院，山东 青岛
关键词: 格点；k-可见；渐近公式；Lattice Points； k-Visibility； Asymptotic Formula
摘要: 设ℤ×ℤ是二维整数格且k,l∈ℕ，若格点(m,n)∈ℤ×ℤ位于形如y=rxk(r∈ℚ)的曲线上，且在(m,n)与原点(0,0)之间的相应曲线段上至多有l−1个格点(不含端点)，则称(m,n)是l-重的k-可见格点。特别地，当重数l=1时，简称(m,n)为k-可见格点。本文给出了方形区域[1,x]×[1,x]中l-重k-可见格点个数的一个渐近公式，这推广了Goins等人关于k-可见格点密度的一个结果。
Abstract: Let ℤ×ℤ be the two dimensional integer lattice and k,l∈ℕ . We say a point (m,n)∈ℤ×ℤ is k-visible with Level-l if it lies on a curve of type y=rxk with r∈ℚ and there are at most l−1 lattice points on the curve segment between points (m,n) and (0,0) (not included). In this paper, we prove an asymptotic formula for the number of lattice points in the square [1,x]×[1,x] which are k-visible with Level-l. This generalizes a result of Goins et al.
文章引用：黄旸. 格点的多重可见性[J]. 理论数学, 2021, 11(2): 271-276. https://doi.org/10.12677/PM.2021.112036

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