Visible lattice points in P\'{o}lya's walk

TL;DR

This paper investigates the distribution of visible lattice points in generalized Pólya's walks on integer lattices, proving the density for perturbed walks and conjecturing the same for twisted walks, supported by numerical experiments.

Contribution

It establishes the almost sure density of visible points in perturbed Pólya's walks and conjectures the same for twisted walks, extending understanding of lattice point distributions.

Findings

Density of visible points in perturbed Pólya's walk is 1/ζ(k).

Numerical experiments suggest the same density for twisted Pólya's walk.

Standard Pólya's walk is a special case covered by the results.

Abstract

In this paper, for any integer $k\geq 2$, we study the distribution of the visible lattice points in certain generalized P\'{o}lya's walk on $\mathbb{Z}^k$: perturbed P\'{o}lya's walk and twisted P\'{o}lya's walk. For the first case, we prove that the density of visible lattice points in a perturbed P\'{o}lya's walk is almost surely $1/\zeta(k)$, where $\zeta(s)$ denotes the Riemann zeta function. A trivial case of our result covers the standard P\'{o}lya's walk. Moreover, we do numerical experiments for the second case, we conjecture that the density is also almost surely $1/\zeta(k)$.