Generalized visibility of lattice points in higher dimensions

Abstract

For any k≥2 and fixed b=(b1,⋯,bk)∈Nk, this paper concerns the number of integer lattice points in Zk which are b-visible from a set of N watch-points. Moreover, for N=2 and b˜=(b˜1,⋯,b˜k)∈Nk with b˜≠b, we consider the mixed visibility, that is, the number of integer lattice points which are b-visible and b˜-visible from two watch-points respectively. We prove asymptotic formulas for the number of corresponding visible lattice points. Part of our results improves a classical result of Rearick.