2022/10/14 by Kui Liu, Liu, Kui, Meijie Lu +3
Mathematics · #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.2210.07464
For any integers k≥ 2, q≥ 1 and any finite set A=\\boldsymbolα1,⋯,\boldsymbolαq\, where \boldsymbolαt=(αt,1,⋯,αt,k)~(1≤ t≤ q) with 0<αt,1,⋯,αt,k<1 and αt,1+⋯+αt,k=1, this paper concerns the visibility of lattice points in the type-A random walk on the lattice ℤk. We show that the proportion of visible lattice points on a random path of the walk is almost surely 1/ζ(k), where ζ(s) is the Riemann zeta-function, and we also consider consecutive visibility of lattice points in the type-A random walk and give the proportion of the corresponding visible steps. Moreover, we find a new phenomenon that visible steps in both of the above cases are not evenly distributed. Our proof relies on tools from probability theory and analytic number theory.