Visibility Of Plane Grid Point

This thesis studies the visibility of lattice points in plane from two problems:one is the b-visibility of lattice points whose coordinates are square-free numbers,and the other is the multiple visibility of lattice points.Let b=(b1,b2)∈ N2 with b1,b2 being coprime,set SN is an admissible set containing N mutually b-visible lattice points.We give an asymptotic formula for the number of lattice points in the region[1,x]2 ∩ N2 N2(x≥2)that are b-visible to the admissible set SN and whose coordinates are square-free numbers.In particular,when N=1,we discuss in detail where the density of lattice points in two-dimensional integer lattice Z2 that are b-visible from one watch-point and whose coordinates are square-free has the maximum value,and where has the minimum value.Furthermore,by giving the discriminates rule of the level of visible lattice points,we obtain an asymptotic formula for the number of lattice points in the region[1,x]2 ∩ N2(x≥2)that are l-level b-visible from the origin(0,0),where l,b∈N.The chapter one mainly reviews the relevant research background of lattice points visibility and the main results of this thesis.The chapter two introduces the basic knowledge involved in the proof of theorems.The chapter three illustrates five different types of visible lattice points:lattice points that are visible from one watch-point along straight lines and curves respectively,lattice points that are jointly visible from several lattice points,visible lattice points with square-free coordinates,and multiple visible lattice points.Finally,in chapter four,we show the proof of the main results of this thesis.