Visible lattice points along curves in higher dimensional random walks

Abstract

Let k≥2 be an integer, and α=(α1,⋯,αk) with 0<α1,⋯,αk<1 and α1+⋯+αk=1, this paper concerns the generalized visibility of lattice points visited by an α-random walk in Zk. Specifically, for b=(b1,⋯,bk)∈Nk with gcd⁡(b1,⋯,bk)=1, we focus on the lattice points in the walk which are b-visible from a set of N watch-points, and we show that, almost surely, the proportion of b-visible steps is ∏p(1−N/ps(b)), where p runs over all primes and s(b):=b1+⋯+bk. This paper generalizes our previous work for the visible lattice points in higher dimensional random walks.